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ECHO RE.OVAL BY DISCRETE GENERALIZEDLINEAR FILTERING
RONALD W. SCHAFER
TSCHNICAL REPORT 466
FEBRUARY 28, 1969
JUN 3
A
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
RESEARCH LABORATORY OF ELECTRONICSCAMBRIDGE, MASSACHUSETTS 02139
Sby th.
CLEARINGHOUSE
Int• dreahd $cp•-nhja{I•,c j & Toh 51
The Research Laboratory of Electronics is an interdepartmentallaboratory in which faculty members and graduate students fromnumerous academic departments conduct research.
The research reported in this document was made possible inpart by support extended the Massachusetts Institute of Tech-nology, Research Laboratory of Electronics, by the JOINT SER-VICES ELECTRONICS PROGRAMS (U.S. Army, U.S. Navy, andU.S. Air Force) under Contract No. DA 28-043-AMC-02536(E).
Reproduction in whole or in part is permitted for any purpose ofthe United States Government.
Qualified requesters may obtain copies of this report from DDC.
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9IST. AVAIL 83d'0 SPECI
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
RESEARCH LABORA.TORY OF ELECTRONICS
Technical Report 466 February 28, 1969
ECHO REMOVAL BY DISCRETE GENERALIZED
LINEAR FILTERING
Ronald W. Schafer
This report is based on a thesis submitted to the Departmentof Electrical Engineering, M. L T., January 19, 1968, in par-tial fulfillment of the requirements for the degree of Doctorof Philosophy.
(Revised manuscript received November 18, 1968)
Abstract
A new approach to separating convolved signals, referred to as homomorphic decon-volution, is presented. The class of systems considered in this report is a member ofa larger class called homomorphic systems, which are characterized by a generalizedprinciple of superposition that is analogous to the principle of superposition for linearsystems.
A detailed analysis based on the z-transform is given for discrete-time systems of
this class. The realization of such systems using a digital computer is also discussedin detail. Such computational realizations are made possible through the application ofhigh-speed Fourier analysis techniques.
As a particular example, the method is applied to the separation of the compo-nents of a convolution in which one of the components is an impulse train. This classof signals is representative of many interesting signal-analysis and signal-processingproblems such as speech analysis and echo removal and detection. It is shown thathomomorphic deconvolution is a useful approach to either removal or detection ofechoes.
F
TABLE OF CONTENTS
L INTRODUCTION 1
1. 1 ;eneralized Superposition 2
1. 2 T•e Cepstrum 6
IH. ANALYSIS OF DISCRETE-TIME HOMOMORPHIC DECONVOLUTION 7
2. 1 Complex Logarithm 8 -
2. 2 Realizations for the Systems D and D-I 12
Z. 3 Integral Relations for the Complex Ceputrum 13
2.4 "Time-Domain" Expressions for the Complex Cepstrum 15
2.5 Complex Cepstrum for Sequences with Rationalz-transforms 17
2.6 Minimum.nPhase and Maximum-Phase Sequences 24
2.7 Exponential Weighting of Sequences 31
2. 8 More General Rational z-Transforms 33
. 9 Examples of Complex Cepstra 36
2.10 Linear Syrstem in the Canonic Representation 42
III. COMPUTATIONAL CONSIDERATIONS IN HOMOMORPHICDECONVOLUTION 45
3.1I Sampled z-Transform. 45
3.2 Fast Fourier Transform 51
3.3 Properties of the Sampled-Phase Curves 52
3.4 An Algorithm for Computing arg [X(k)] 54
3.5 Other Computational Considerations 59
3.6 Computation Time Requirements 61
3.7 Minimum-Phase Computations 62
3.8 Sampling of Continuous-Time Signals 65
IV. ANALYSIS OF SPEECH WAVEFORMS 684. 1 Speech Production and the Speech Waveform 684. 2 Short-Time Transform 70
:4.3 Short-Time Complex Cepstrum of Speech 71
4.4 Examples 73
V. APPLICATIONS TO ECHO REMOVAL AND DETECTION 77
5.1 A Simple Example 77
5. 2 Complex Cepstrum of an Impulse Train 83
5.3 Distorted Echoes 9!
5.4 Linear Systems for Echo Removal and Detection 95
5. 5 Short-Time Echo Removal 101
5.6 Removal of Echoes from Speech Signals 110
5.7 Effect of Additive Noise 114
5.8 Detection of Echoes 118
iii .i.
CONTENTS
VI. CONCLUSION 120
6. 1 Summary 1206. 2 Suggestions for Future Research 120
Appendix Vector Space for Convolution 121
Acknowledgment 124
References 125
iv
AJ
r1. INTRODUCTION
In many physical situations, we encounter signals or waveforms that may be repre-
sented as the convolution of two or more components. One c.. so of these problems
arises when a signal is distorted by transmission through a linear system. For example,
fthe effects of multipath and reverberation may be modeled in terms of a signal that is
Spassed through a linear system whose impulse response is an impulse train. In this case
we may be interested either in recovering the undistorted signal or in determining the
parameters of the impulse response. A similar class of problems arises when we are
given a waveform that can be representcd as a convolution of two or more component
signals, and we may wish to determine these components so as to characterize the wave-
form or the physical process from which it originated. I. --- example, certain segments
of speech waveforms may be represented as the convolution of sev-ral components.
Most speech bandwidin-compression schemes are based on the dterz.Jination of the
parameters of these component waveforms.
The process of separa'ing the components of a convolutirn is termed deconvolution.
n performing deconvoution of a wavelorm we must ietermine an appropriate transfor-
mation of the waveform into the desired component waveform. A common method ofS~deconvolution is ca,•lled inverse filtering. lit this method, the signal is transformed by
a linear time-invariant systern whose system function is the reciprocal of the Fouriertransform of the components to be removed. Although inverse filteri.g has been suc-
cessflly applied in processing many different types of signals, it is limited by the
necessity of knowing the signal to be removed, as well as having a sensitivity to additive
noise. Another deconvolution technique is based on the Wiener theory of linear filtering.
This technique has been extensively applied in processing seismic waveformvs.6 In detec-tion of echoes, maximum.n-likelihood methods 8 and correlation have been used. Variour
other techniques have been developed for special situations. 4 ',7 It is difficult to compare
the various methods of deconvolution because generally each metnod requires different
information aboit the signals and the objectives of each method are not precisely thesame. Nevertheless, it is clear that there is not a single best method that can be applied
to all deconrolution problems. Given the importance of the problem of deconvolution,
it seems that even though a variety of methods are available, at present, it is cogent to
investigate othzr approaches, The detailed consideration of a new api roach to deconvo-
lution is therefore the subject of this report.
The approach to deconvolution presented here was originally proposed by Professor
Alan V. Opp.mnheirn as art application of the theory of generalized superposition. 1 3 The
parallel development of the applications of this technique to speech analysis 1 9 ' 2 0 byOppenheim, and to echo removal9 by the author led to the theoretical formulation of the
technique presented in this report.
Our purpose is to give a detailed discussion of the characteristics of this new
approach to separating convolved signals. Since it appears that digital realizations of
this signal-processing method are most promising, oi'r analysis will be confined to
discrete-time signals and will be based on the z-transform, We shall also investigate
carefully the actual realization of technique in the form of algorithms for a digital com-
puter. As an example cf the use of this technique, we have considered the problem of
deconvolution for the class of signals that are represented as the convolution of one or
more waveforms with an impulse train. This kind of representation is characteristic
of the waveforms of speech and music and many other acoustic disturbances. Also,
seismic signals, sonar signals, and many biological signals are in this class. In fact,
any signal that is quasi-periodic by nature, or any signal that has been transmitted
through a reverberant environment will have such a representation.
We shall now review the theory of generalized superposition, its relation to
"cepttral" analysis,10-13 and its application to deconvolution. In Section II a detailed
analysis of the technique will be presented, and in Section III we shall focus on compu-
tational considerations. In the rest of the report we shall discuss applications to speech
processing and to echo removal and detection.
1. 1 GENERALIZED SUPERPOSITION
A system is often defined abstractly as a unique transformation of an input signal
or waveform x into an output signal y. The signals are represented by functions of
time, and the system corresponds to the mathematical concept of an operator. Such
transformations are denoted by
y = T[x].
In order to characterize and classify systems, we place restrictions on the form
of the operator T[ ]. For example, the class of linear systems is characterized by the
property
T[ex +bx2 ] = aT[x1 ] + bT[x2 ]. (1)
Similnarly, the class of time-invariant systems is characterized by the property that if
T[x(t)] = y(t),
then
T[x(t+to)] = y(t+t0 ). (2)
ThE, class of linear time-invariant (LTI) systems has both of the properties
expressed by Eqs. 1 and 2. As a direct consequence of these properties, it can be
shown27 that all LTI systems are described by the convolution integral
y) X(T) hit-T) d" h(T) x(t-T) dT, (3)
where y(t) is the output, x(t) is the input, and h(t) is the response of the system
2
to a unit impulse. The class of LTI systems is very Lmportant for three basic reasons.
1. Linear time-invariant systems are rather easy to analyze and characterize.2. It is possible to design linear systems to perform a large variety of useful
functions.
3. Many naturally occurring phenomena are accurately modeled using lin:ar sys-tem theory.
The first of these comments is primarily a consequence of the principle of super-position (Eq. 1) which character, izes linear systems. In particular, when .he input
is a sum of component signals, a linear system is very conveniant for separatingone component from the other. As we shall see, our approach to deconvolution is
motivated by similar considerations.
Classes of systems are defined by placing restrictions on the transfo-malioa thatrepresents the system. To state that a system is nonlinear does nothing tv characte ize
the properties of that system. An approach to characterizing nonlinear systems whichis based on linear algebra has been presented b,.- Oppenheim.I In this approach it isrecognized that vector spaces of time functions at the input ard output of a system
can be constructed with a variety of defiiriti- iý. of vector addition and scalar multipli-cation. Thu3 many nonlinear systems can "',i r ýp. zented as linear transformationsbetween vector spaces and can thus be sairi: 4o •,*,(y a generalized principle of super-
position. Nonlinear systems of this type have beEn called homomorphic systems toemphasize the fact that they are represe-,s.0 by algebraically linear transformations.If we take the operations of vector addition to be the same in the input and output spaces,
then a generalization of the linear filtering problem follows. This approach applied toSthe separa~ion of convolved signals is appropriately termed homomorphic deconvolutiono
0 KYHx xl10x2 y =H[x] :
= H[x,1 40H [Y2]
Fig. 1. Representation of a homomorphic system that obeys ageneralized principle of superposLtion for convolution.
Tae class of homomorphic systems of interest for deconvolution is one in whichvector addition i3 defined as convolution. A system of this class is shown in Fig. 1.The system H is characterized by the fact that if
H[xl] - y1 and H[x 2] - y2 ,
b, en
3
H[ (a)x (b)x, - 9H~ ( 9) {b)H - (a)y (b)y2 (4)
where 0 denotes convolation, and (a) denotes scalar multiplication. (The meaning of
scalar multiplication is discuised in the Appendix.) Comparison of ECIs. 1 and 4 should
suffice to show why we use the term "generalized superposition." it has been shown1
that all homomorphic systems have a canonic representation as the cascade of a non-
linear system followed by a linear system and then another nonlinear system. For con-
volutional input and output spaces, this canonic form is shown in Fig. 2. The system D
is a homomorphic transformatior from a convolutional space to an additive space so that
if D[xl] = X, and D[x2 ] x thex
[(a)x 1 Axj
D[ 2 =aDx,,- bDrlx =a1 + b~x2 '
The system L is a linear system in Uie conventional sense so that if
,= and T [ =2]
then
L[axl+bX2 ] = aL[^1 ] + bV[- 2I = ay +ay-12'
The system D is the inverse of the system D and it serves to transform from the
additive space of L back to the convolutio,,al space.
I- D LD-----1---------X Y
H
Fig. 2. Canonic. form for homomorphic deconvolution.
The canonic representation is extremely important. All homomo.-phic systems with
convolution for both input and output operations have the same form and differ only in
the linear part, L. This is the reason for referring to Fig. 2 as a canonic representa-
tion. It should be clear that such a representation allows us to study such systems by
first focusing our attention on the system D, and then applying the well-developed tech-
niques of linear system theory to aid in understanding a particular over-all system H.
For example, if we are interested in designing a homomorphic system for recovering
signal xI from the convolution x . xI x2 , we need to choose the system L so that X2
is removed from the additive combination existing at the output of D.
The system D depends entirely on the specific operation for combining .ignals at
4
the input and thus is the same for all homomorphic systems for deconvolution. For
this reason, the system D is called the characteristic system 'or homomorphic
deconvolution.
The nature of the transformation D is suggested by considering the Fourier trans-
forms of x and 2. Suppose
x =x 1 8 x229
so that the Fourier transform of x is
X = X1 * X2 , (5)
where X, X1, andX 2 are the Fourier transforms of x, x, , and x. We see also thatthe Fourier transform of 2 must be of the form (
whrX = X I + X2 a (6)!A A
Equations 5 and 6 suggest that under an appropriate definition of the logarithm, we might
define the system D to be the system whose output Fourier transform is the complex
logarithm of the transform of the input; that is,
X = log IXI. jFurthermore, this suggests the method of realizing the transformation D shown in
Fig. 3.
Thus homomorphic deconvolution is based on transforming a convolution into a sum
and then using a linear system to separate the additive components. The result is then
transformed back to the original input space.
ii+
FOURIERINVERSE +FOURIER COMPLEX FOURIER
TRANSFORM LOGARITHM X = log I TRANSFORM IE
L -J
D
Fig. 3. Formal realization of the characteristic systemfor homomorphic deconvolution.
We have chosen for investigation, as examples of the application of homomorphic
deconvolution, the class of signals t.hat can be represented as a convolution in which
one of the components is an impulse train. As an example of this class consider
x(t) = s(t) + as(t-t0 ) = [uo(t)+auo(t-to)] ® s(t).
The Fourier transform of this equation is
5
X(Q)) r SM -jwt0 l.
The complex logarithm is formally
X'= log [S(w)] + log (I+ajwt )
We note that the second term in this expression is periodic in w with a repetition rate
proportional to t
Suppose we view log [X(W)] as a waveform to be filtered with a linear system. We
note that if the spectra of log [S(w)] and log I + j0 do not overlap, the separation
ofcomponents is relatively easy. Alternatively, we require that the term
log 1+ aeo) vary rapidly, compared with the variations in log [S(w)]. Thus we see
that the transformation D allows us to transform a convolution of waveforms into a sum
that, under appropriate conditions, can be separated by a linear system. This allows one
who is familiar with linear system theory to apply all of his experience and intuition to
this technique of deconvolution simply by focusing his attention on the log of th'. Fourier
transform and interchanging the roles of time and frequency.
1.2 THE CEPSTRUM
Independently of Oppenheim's formulation of the theory of homomorphic systems and
our subsequent work, Bogert, Healy, and Tukey10 recognized that the logarithm of the
power spectrum ithe Fourier transform of the autocorrelation function) for a signal con-
taining an echo should have a periodic component whose repetition rate is related to the
echo delay. Thus the power spectrum of the logarithm of the power spectrum should
exhibit a peak at the echo delay time. This function was called the "cepstrum" by trans-
posing some letters of the word "spectrum." Noll 1 2 has traced the evolution of cepstrai
analysis and also discussed various definitions of the cepstrum which have been
employed. Although cepstral methods have been developed from an empirical point of
view, we can see that the cepstrur. is clearly related to homomorphic deconvolution. The
basic difference is that we shall employ a Fourier transform (magnitude and phase),
rather than the power or the energy spectrum. We do this because we are concerned
with the more general problem of recovery of signals as opposed to detection of echoes.
To emphasize this distinction, we shall refer to the outpat of the characteristic sys-
tem D as the complex cepstr1in.
6
I1. ANALYSIS OF DISCRETE-TIME HOMOMORPHIC DECONVOLUTION
We have introduced the concept of systems that obey a generalized principle of super-
position in which addition is replaced by convolution. Since it appears, at present, that
such systems can be most easily realized digitally, we shall be concerned henceforth onlywith discrete-tirre systems of this class. Thus our signal vectors are sequences ofnumbers, and convolution is defined as
00
x(n) = I_, xi(k) x2 (n-k). (7)k= -oo
The canonic form for discrete-time homomorphic systems is shown in Fig. 4, where
x is the input sequence, and x is the complex cepstrum. The system D characterizes
all systems of this class. Therefore we shall begin our study of such systems with a
s'udy of the 3ystem D, and then consider the choice of the linear system L.
+ + + + ID -. L D-
I 1
H
Fig. 4. Canonic form for discrete-time homomorphic deconvolution.
The properties of the transformation D can best be analyzed by considering the z-
transforms of x and x .'26 If x is a convolution,
x=xI 6 x2f
then
X(z} = XlI(z) • X 2 z). (8)
(Note that O denotes discrete-time convolution as in Eq. 7.) We require that if x is a
convolution as in Eq. 7, then
A A AX=xI +x 2 .
Thus the z-transform of 2 must be of the form 0
X(z) = xI(z) + X2 (z). (9)
If we compare (8) and (9), we see that the requireme'it is that the system D effectively
7
transform a product of z-transforms into a sum of corresponding z-transforms. We
shall show that, under appropriate definition of the complex logarithm,
log [X WZ)X2 (z)] log [XI(z)j + log [X2 (z)].
Thus we are led to define the system D as one for which the z-transform of the output
is the complex logarithm of the z-transform of the input. That is,
A noX(z) = (n) z- - log [X(z)]. (10)
n= -oo
Since log [X(z)] must be a z-transform, it must have the properties of a z-transform.
In particular, we must be able to define a region (actually a Riemann surface) in which
log [X(z)] is single-valued and analytic and possesses a Laurent series expansion. Thus
before proceeding to the actual definition and discussion of the realization of the sys-
tem D, it is first appropriate to review some of the properties of the complex
logarithm.
2.1 COMPLEX LOGARITHM
The function X(z) can be expressed as
xjz) - arg [X(z)]
The logarithm of X(z) is defined as
log [X(z)] = log IX(z) + j arg [X(z). (0 1)
since e 2irq = I for any positive or negative integer q, it is clear that we may always
write arg [X(z)] as
arg [X(z)] = ARG [X(z)] * j2nq,
where q = 0, 1, 2 ... , and
-w < ARG [X(z)] -< V.
Therefore log [X(z)] may be expressed as
log IX(z)] = log IX(z)I + j ARG [X(z) * j 2rq. (12)
That is, the complex logarithm is multivalued, with infinitely many possible values. The
principal value of log [X(z)] is defined as the value of Eq. 12 when q = 0, and ARG [X(z)]
is called the principal value of arg [X(z)). (Henceforth, the principal value of an angle
will be denoted by capital letters.)
The transformation D must be unique. Therefore the logarithm must be so defined
8
I that there is no ambiguity with respect to its imaginary part. Furthermore, we require
that log [X(z)] be analytic in some annular region of the z plane because the values ofA
the complex cepstrum x are defined as
A •C]n-1X{n) =•- log [X(z) ]z d,. (13)
In Eq. 13, C is a circular contour specified by
z = eaj -ir < w -< ir,
where ea is the radius of the circle. In Eq. 13 it is assumed that log [X(z)] has a
Laurent series expansion as in (10). Thus we must insure that log [X(eO+JW)] is analytic
in an annular region containing the circle with radius eu. This region is appropriately
called the region of convergence of log [X(z)] or of X(z).In general, the principal value of the phase, ARG [X(eO+JO)] will be a discontinuous
function of w. In fact, ARG [X(eO+jO)] will be discontinuous for values of w for which
arg [X(eI+J = nw, n = *1,+3,+5,
A typical example of a phase curve and its corresponding principal value is shown in
Fig. 5. If the principal value of the phase is used in defining the complex logarithm,
Org NOV Jo)]
-2w
-3w
-4v
ARG [XMeo+J))
(b)
Fig. 5. (a) Typical phase curve for a z-transform evaluated on acircular contour about z = 0.
(b) The principal value of the phase curve in (a).
?9
its derivative does not exist at the points of discontinuity of ARG [X(e7+J4)]. Therefore
the function log [X(ec+Jo)] would fail to be analytic at these points. Becaube log [X(z)]
must be analytic on the contour C, we must eliminate such singular behavior by corn-
puting a phase curve with no discontintuies.
We also recall that if
X(z) = XI(z) X2 (z),1
then we require that
log [X(Z)] = log [X1(z)] + log [X (z)],
on the contour C.
If we write
Xl(Z) = JXl(Z)I ej arg [X(z)]
and
X2 (z) = 1X2 (z)I ej arg [X(z)J
then we require that
log IX(z)l = log IX(z) I + log IX 2 (z)I (14)
and
arg (X(z)] = arg [XI(z)] + arg [Xz(z)], (15)
where z = eu+jw and -w < w 4 7r. Since log IX(z) is simply the logarithm of a positive
real number, (14) will be satisfied whenever I X (z) and X2 (z) are nonzero and finite.
With respect to the phase angles, we can write
arg [X(z)] = ARG [X(z)] * j2irq (1 6a)
arg (XI(z)J = ARG [XI(z)] ( j2wqI (16b)
arg [X 2 (z)] = ARG [X2 (z)] * j2vq 2 , (1 6c)
where q, ql, and q 2 are integers. Clearly, (15) will hold only if we choose the appro-
priate value for arg [X(z)]. For example, suppose that we choose the principal value
for all angles. It can be shown that, in general,
ARG (X(z)] * ARG [XI(z)] + ARG [Xz(z)].
One way of insuring that (15) will always hold is to assume that all angles are computed
so that they are continuous functions of o as z varies along the contour C specified by
z =ec+jw. Chis implies that for each value of w, we have chosen the proper values for
10
A
q, q1 , and q2 in Eqs. 16 so that all angles are continuous functions of W. In the actualcomputation we only compute arg (X(z)], so the proper choice of q, and q. is implicit
in the proper choice of q. Thus requiring that the phase curve be continuous alsoimplies that Eq. 15 is satisfied.
Two other restrictions on the form of arg [X(z)] result from considerations that do
not have to do with the logarithmic operation. If we require that x(n) be real when x(n)
is real, the real part of X(e ) must be an even function of w and the imaginary partof X~e+jt) must be an odd function of w. Since X(eU+Jt)) is even for real x(n), so is
Re [X(e'+Jw)] = log IX(e'+Jt)I.
The requirement on the imaginary part implies that we must define
arg [X(ea+Jw)] = -arg [X(eo-Ju))].
A final condition is required because log [X(z)] is to be the z-transform of the sequence
x; log [X(e7+j a)] must be periodic in w with period 2w. That is,
log IX(eo+Jto) = log IX(e'+jwAj~k)I
and
arg [X(eo+Jwa)] = arg [X(e +jwjZwk)],
where k = 0, 1, 2..... This periodicity and the even and odd symmetry propertiesimply that log IX(e'+ja) I has even symmetry about w 0, *7T, *21r, ... and likewise
arg [X(e'+Jw)] has odd symmetry about w = 0, *ir, i21,.
To summarize, the conditions that are imposed on
Im [X(z)] = arg [X(z)]
are the following.
(C1) arg [X(z)] is a continuous function of w for z = e +j4.
(C2) arg (X(z)] is an odd function of w for z = e
(C3) arg (X(z)] is periodic in w, with period 2w for z = eo+j6).
Conditions similar to (C2) and (C3) apply to log IX(z) and follow automatically from
the definition of the logarithm of a real number and the symmetry properties of the mag-nitude of a z-transform. These conditions are the following. "
(C5) log JX(z) j is an even function of wa for z = ea+jt.
(C6) log X(z) is periodic in w, with period 2: for z ey+j3w.
11 1.
2. 2 REALIZATIONS FOR THE SYSTEMS D AND D-!
We have seen that if special care is taken in defining the complex logarithm, the
logarithm of a product of z-transforms is the sum of the logarithms. Furthermore,
under these conditions, log [X(z)] can also be thought of as the z-transform of the
I-ASFOm log I I z-TRANSFORMX X(z X(Z) = log [X(z)JX
D
Fig. 6. Realization of the characteristic system for homomorphicdeconvolution using z-transforms.
complex cepstrum. Thus, one realization of the system D is that shown in Fig. 6.
The complex cepstrum is seen to be the result of the equations
X(z) = cc x(n) (17a)n= -oo
X(z) = log [X(z)] (17b)
x(n) = log [X(z) z dz, (17c)
where the closed contour C lies in a region in which log (X(z)] has been defined as
single-valued and analytic.
TWO-SIDED exp INVERSE
z -TRANSFORM 9 (z) Y(z) = exp Cy(z) z
y II
Fig. 7. Realization of the inverse characteristic system for homomorphicdeconvolution using the z-transform.
Similarly the inverse of the system D is shown in Fig. 7. Thus, we obtain for the
output of D-I the equations
12
n= -- nA
Y(z) = exp[Y(z)] (1 8b)
y(n) = 1 , Y(z) zn ldz. (18c)
In (1Bc) the contour C' must be a closed contour in the region of convergence of the input
z-transform X(z). This is required because if the linear system is the identity system,
we require that the over-all system be the identity system; that is, if
yn -- (n),
then
y(n) = x(n).
2.3 INTEGRAL RELATIONS FOR THE COMPLEX CEPSTRUM
We 'xave shown that the complex cepstrum can be obtained from the set of equations
G0
X(z) x(n) z (1 9a)
n= -0o
X(z) log [X(z)] = log IX(z) + j arg (X(z)] (1 9b)
x(n) =- X(Z) z dz. (I19c)
These equations constitute a definition of the systewt D and also lead to a computationalrealization. We shall consider Eq. 19c and show how it may be used in studying the prop-
erties of the system D.
We have seen that the circular contour C must lie in the region of convergence of
AAX(z). In using these equations for computation, part of the definition of the system D
is the choice of the region of convergence of X(z) In general, the two-sided transforms
X(z) and X(z) have regions of convergence which are annular regions of the z plane. 2 3 ' 26For example, we shall usually denote the region of convergence by A relation of the form
R+< jzj <R _.
By definition, these regions can contain no singularities of the z-transforms. The regions
of convergence may, however, contain zeros of the z-transforms, and we shall see thatA Athese cases require special handling. Since X(z) is the comrlex logarithm of X(z), X(z)
will have singularities at all of the singularities and at all of the zeros of X(z). Similarly,
13
AI ,
X(z) will have zeros at all of the ones (X(z) = ej0 ) of X(z). Therefore we see that the
region of convergence of X(z) can be the same an the region of convergence of X(z) only
if X(z) has no zeros in its region of convergence. On the other hand, it should be clear
that we are free to choose any annular region that does not contain singularities or zeros
of X(z) as the region of convergence of X(z).
The choice of the region of convergence for X(z) is based primarily on computational
considerations, and at least two differ'pnt choices have been found useful. In any case,
it should be clear that for a given input sequence x, it is possible to obtain many dif-
ferent output sequences x depending on the region of convergence that is chosen forAX(z). This does not mean that the output is not unique because the choice of the contour C(and therefore the choice of the region of convergence of X^(z)) is part of the definition
of the characteristic system D. Once this contour is fixed, the output is uniquely deter-
mined. t
Let us temporarily leave the contour C unspecified and obtain a more useful expres-
sion for 'X(n). Using Eqs. 19b and 19c, we obtain
x(n) =-jl- log 1X(z)]Zn- dz. (20)itj C
eO+j•,If we note that the contour C is specified by z =e with -7 < w < ff, we can
write (20)
X(n) = -, log [X(eY+JWj)] ean eJwn dw. (21)-iT
We shall proceed to integrate (21) by parts under die assumption that log [X(eo÷J)
is a single-valued periodic function of w which is everywhere continuous. We shall find
that this assumption is somewhat restrictive, but we shall also show how the results
derived here apply to more general circumstances.
If we integrate (21) by parts, we obtain, for n # 0,
"(n) [log - 2•ejnl Cr= -- log [X(e7+jw)] e eJ(nZnjn " oX(e+ Jw)]epn] Zjn .
Because both log [X(ef+J1w)] and ejwn are periodic with period 2w, the first term in the
expression above vanishes. Since we have assumed that log [X(e0'+j3)] is continuous
everywhere, we obtain
d al~ +jw)
_n - e en e (22)2(n)= 2jn ' ~X(e'+W
Since the logarithmic derivative is also analytic in the region of convergence of X(z) we
may write (22)
14
-(n) - zX'(z) zn 1 dz, (23)
where the prime indicates differentiation with respect to z. The contour C is, ofA
course, still in the region of convergence of X(z).
The value of x at n = 0 is obtained directly from Eq. 21; that is,
^() x. D log [X(e0+j()] di.
Since arg [X1eU+J¶)] is an odd function of w and log JX(e"+Jo)) is an even function of u,
=(O) log JX(e+Jw)J dc6. (24)
Thus as an alternative to Eqs. 19 for analysis, and possibly for c,)mputational pur-
poses, we have Eqs. 23 and 24, under the assumption that X(e'+jw) is a single-valued
and coutinuous function of w for all w. We shall see that this condition must be relaxed
in order to include most situations of interest. (This will be done in section 2. 9.)
2.4 "TIME-DOMAIN" EXPRESSIONS FOR THE COMPLEX
CEPSTRUM
The expressions just derived gave the complex cepstrum x explicitly in terms of
the z-transform of x. Equation 23 may be used to obtain an implicit expression in
terms of x(n) &nd /(n) which, in certain cases, reduces to a recursion formula.
This implicit relation may be derived as follows. If we assume that log [X~eg+Jw)]is continuous for all w, we can write
/A X' (z)X' (Z) = --X(z)
Rearranging this expression, we obtainA
zX'(z) = zX'(z) X(z). (25)
Since
0o
X'(z) =z 1 , nx(n) zn
we see that the inverse z-transform of (25) is
nx(n) = . k2(k) x(n-k). (26)
k= -oo
15
p
Ak
There are several special cases of (26) that are worthy of special consideration.
Case l: x(n) = 0 for n < 0, and x(O) 0.
In this case we can write
n
nx(n)= , k2(k) x(n-k),k=.-oo
which can be written as
x~) n-I x(n-k)(n=x(n) = I I k2(k), n 0 . (27)
x(0) n k=-cj' x(0)
Thus we see that AX(n) depends on all values of x and the values of 2 for k < n.
Case 2: Suppose that x(O) * 0 and x(n) = 0 for n < 0. If we further assume that x(n) = 0
for n < 0, we obtain froir Eq. 27
n-IA x(n) n- x(n-k)
x (n) ) x(k) n>0. (28)
k=0x(O)The value of 2(0) for sequences of this type can be shown to be (et. section 2. 5)
2(0) = log x(O). (29)
Requiring that 2(n) = 0 for n < 0 is equivalent to choosing the contour C in (23) so as to
enclose all of the poles and zeros of X(z). If X(z) has poles or zeros outside the unit
circle, it can be shown2 3 that 2 (n) will be unbounded for large n, since we are effec-
tively choosing the region of convergence to be outside of all of the poles and zeros of
X(z). This will not be the case, however, if X(z) has all of its poles and zeros inside
the unit circle.
Thus when X(z) has all ui its po)es and zeros inside the unit circle, 2(n) satisfies a
recursion relation that could be us,; r in actually computing X(n). (Discussion of the util-
ity of this expression is reserved for Section In.)
Finally, we observe that Eqs. 28 and 29 provide a way of obtaining x from x, that
is, a recursive reltion for the inverse characteristic system. By rearranging Eqs. 28
and 29, we obtain
x(0) = e"(0) (30a)
n-I
x(n) = x(n) x(0) + ~~ 20(k) x(n-k) n > 0. (30b)k=0
Equations 30 represent a realiza-ion of the inverse characteristic system for sequences
whose z-transforms have no poles and zeros outside the unit circle.
16
.......
Case 3: Suppose thatxO) * 0, x(n) = 0 forn<0, andx(n) 0 forn<0 andn>M.
In this case, Eqs. 28 and 29 take the form
X(0) = log x(O) (31 a)
x(n) n-1 i k x(n-k)""n - / -- 0 x 0 < n < M (31b)
x(O) kO x(O)
x~) n-l r.(n-k)x Qn.) (k ) n>M. (31c)
x(O) k=n-M
Case 4: Letx(n) = 2 (n) = 0 forn>Oandx(O)* 0.
These assumptions are equivalent to taking the contour C in (23) to be inside all ofA
the poles and zeros of X(z). Thus for a stable sequence x, we require that all of the
poles and zeros be outside the unit circle.2 1 Using Eq. 26, we arrive at
X(0) log x(0) (32a)
x(n) 0x(n) = - x(k) x(n-k) n < 0. (32b)
x(0) k=n+ 1
2.,5 COMPLEX CEPSTRUM FOR SEQUENCES WITH RATIONAL
z-TRANSFORMS
In actual computations, we are always restricted to sequences of finite length and
hence to z-transforms that are simply polynomials in z Thi-3 it is not a significant
restriction if we consider z-transforms of the form
1/(1-a kz-l 11( bkZ
X(z) = A k=I k=l (33)Pi IPo
n (l-ckz-)1 (l-dkz)k= I k=1
where A is a positive-real constant, and the ak, bk, ck and dk are nonzero complex
numbers whose magnitudes are less than one, If x is a real sequence, then the ak. bk,
ck and dk occur in complex conjugate pairs. Careful examination of Eq. 33 shows that
there are mi zeros and pi poles inside the unit circle, and mo zeros and p0 poles out-
side the unit circle. Clearly, (33) is not tWe most general rational z-transform, since
A could be negative and in general we must include a factor of the form zr to account
for all shifted versions of the sequence x. Since our method in computation will be to
deal with these issues separately, we shall defer discussion of these points.
17
.•'•• • • •'•" ¢a• 4&.9
(1-z~1 )
w/2
zo -w/2 ,- 2 w wdI
-l-
W112 2w w
w/
w(2w)
0T2
Fi.8zhs ýre o eo nieteui ice
a1
w/*2
0/2 -w/2 2iu
it (a)
W/2
l/z w -, 2, U
(b)
0 (c) -w/2 2
00
1/z 0
Fig. 9. Phav curves for zeros outp^ýde the unit circle.
19
We have shown that the phase curve must be a continuous function of w. Since
arg [x(ea÷j')] will w! the sum of the arguments of each multiplicative factor in Eq. 33,
it is helpful to consider the contributions from each of these factors. Figures 8 and 9
show the typical pole-zero plots and one period of phase curves when z = e j for each
type of numerator factor in Eq. 33. The corresponding denominator factors produce
phase curves that are the same except for sign. In all cases, the peak value of these
phase curves is less than or equal to n/2. The value w/2 is attained only when the zeros
(or poles) lie on the unit circle. If the zeros (or poles) are on the unit circle, the phase
',urves become discontinuous. We also observe from Figs. 8 and 9 that all of the phase
curves of these factors are zero at w = 0, iir, +2w .
Since the total phase curve for Eq. 33 is the sum of the phase curves of each factor,
the total phase curve wit'. be zero at w = 0, in, 2n ...... Furthermore, it is clear that
arg [X~eg+Jo)] will in general be greater in magnitude than w. Therefore in computing
the phase, we must use an algorithm that enables us to determine the correct phase
curve, that is, one without discontinuities.
One such algorithm computes the principal value of the phase and then determines
the correct multiple of 2w to add to or subtract from the principal value for each value
of w. This algorithm is discussed in Section III.
We are also interested in log I X(e'+Jw) 1, since this is the real part of the complex
logarithm. Since the magnitude is an even function of (, it will have the same general
form for poles and zeros both inside and outside the unit circle. Let us consider a fac-) ej4tor such as (I -Z 0 Z-) for z = eJ , and zO = Izol e The magnitude of such a factor
is
I-z 0 e-3 " = I +Izo0
2 -21zoIcos (-o)J1
Taking the logarithm, we obtain
log I l z 0 e-JW =-Flog +I _212z0Jcos (W-ýO,]
This function is sketched in Fig. 10. We note that it is periodic with period 2w. The
maximum positive value is - log (I+ 2 1 z . IzO12) which approaches log (2) as IZo Iapproaches 1. Similarly, the most negative value is 1 log (I -2 zo I + I o 2 which
approaches log (0) or -oo as I zoI approerhes i.
Since log I X(eJ')I is the sum of terms such as this (with negative signs for denom-
inator factors), we would expect that log I X(eJW)j would have an appearance similar to
that of Fig. 10, except that in general there will be peaks corresponding to each of the
poles and zeros of X(e JW). A typical example of log IX(eJw) I is showr in Fig. 11.
We have seen that z-transforms having the form of Eq. 33 satisfy the requirement
that log [X(e'+J0)] be continuous everywhere. Thus we may employ Eq. 23 to evaluate
the complex cepstrum. The integrand zX'(z)/X(z), in this case, is
20
Ii
log l-l z eIs % 'i- l
2/2 lg (I(21z2 +Io z2)
Fig. 10. Logarithm of the magnitude of the complex factor (I -z 0 e-jw)
for 0 Jz0 e00o.= Izol 4.
log IX(OjI"1
I• IxIh•
Fig. 11. Typical curve for the logarithm of the magnitude of the z-transformof a finite-length sequence.
21
mI - 'o Pi -m1 POk z) = k __b___ I k __+ dkZZ I I akZ 1 bkz -1 - .dz (34)
= k= I k=1 Ck k=1
•()=- CzX'l(z) z n-1 dz,
we see that if we desire a stable sequence (one whose values approach zero for large n),
we must choose the region of convergence to include the unit circle. Each factor in
(34) is the z-transform of an exponential sequence. Therefore, if the contour is taken Ias the unit circle, A(n) is given by
Pin m. nx~n)= I Ck akn> 3an n
" - - n >•1 (35a)
k=1 k=1
P0 -b;nn o dkn n,-I. (35b)k= I k= I
The value of ^(O) is obtained from Eq. 24 with a = 0. Therefore
X(o) -- log I X(eJ') I dw.
Each factor of X(ej3 ) I has tne form
1 -a e*Jjl = 1 + la12 - 21aI cos (wF arg[a]),
and it can be shown that
2- log (l+jaI2 -2jaf cos(wFarg[a])) dw= O,--n
if jaj 4 1. Therefore we see that
x(0)= -. log A dw = log A. (36) j
Equations 35 and 36 express 2 (n) in terms of the poles and zeros of the rational
z-transform. They also illustrate an important property of the complex cepstrum. It
is clear from Eqs. 35 that
22
JIx(n)•14 B a' n *0,Inl
where B is a positive constant and a is the magnitude of the pole or zero that is closest
to the unit circle. IIn many simple cases, it is not necessary or desirable to use the integral formulas
for purposes of analysis. This is particularly true when the z-transform is a rational
function. In this case a power-series expansion of log [X(z)] is usually more convenient.
Under the assumptions that log [X(z)] is defined to be single-valued and analytic inthe region of convergence, and that X(z) has the form of Eq. 33, we may write
m.1
log [X(z)]= log A + k leg (l-akZ-k= 1
in 0 imo Pi O
+ log (1-bkz) - log (1-c ) Z-1 log (Il-dk) (37)k= Ik= I k= I
A
Since we define X(z) as a z-trauaform, it must be true that
0o
X(z) = log [X(z)] = I (n) z- (38)
n= -go
Thus we immediately see that
x(0) = log A.
If we effectively take the contour C to be the unit circle, then each of the remaining
terms in (37) can be expanded in a Laurent series about z = 0. For example, we can write
00
log I -akz') = - I q ~ for Izi > lakn- I
-log (I-Ckz-)= Z_ ckz-n for Izi > Icki
n= 1
-1 -n
log (l-bkz) = -"n- - for jzj < lbk I.
-223
_ -n I l < I-log (I -d Z) -- d -n for <Id
n---go
Therefore if we add these convergent series and collect the coefficients of z-n, we can
determine £(n). In general, we see that 5(n) can be written
x(O) = log A (39a)
Pi n mi
k= I k=1
n -=. (39c)k= 1 k=!
Equations 39 agree with Eqs. 35 and 36, as we would expect. The real value of the
power-series approach is best illustrated by our use of it in discussing echo removal
applications in Section V.
2.6 MINIMUM-PHASE AND MAXIMUM-PHASE SEQUENCES
We have considered a realization of the system D which was based on the z-
transform. In some cases it is possible to take advantage of the properties of the z-
transform to obtain simplified results. For example, we have seen (section 2.4) that
under certain conditions, x(n) obeys a recursion formula. We shall now consider these
cases in detail and present an alternative computation scheme.
A minimum-phase sequence is defined as a sequence whose z-transform has no poles
or zeros outside the unit circle. Furthermore, the region of convergence for the
z-transform includes the unit circle. For rational z-.transforms, X(z) is of the form
mi
X(z) = A k=1Pi
11 (0-ckz-)k=1
where the ak and ck are complex numbers whose magnitudes are less than one, and the
region of convergence is specified by
Izi >max ICkk1
k
Such sequences have the properties
x(n) 0 n < 0 (40a)
x(O) * 0 (40b)
24
00 x(n)! < Co. (40c)
n=O
We have shown that the complex cepstrum of such a sequence has the properties
x(n) = o n < 0 (41a)
*x(O) = log x(O) (41b)
00 IQ(n)I < oo. (41c)
n=O
Since Eqs. 40 and 41 are necessary and sufficient conditions, these equations could be
taken as the definition of a minimum-phase sequence.
An entirely analogous situation is called maximum-phase. In this case, X(z) has
all poles and zeros outside of the unit circle, and the region of convergence includes
the unit circle. In this case, x has the properties
x(n) 0 n > 0 (42a)
x(0) # 0 (42b)
x(n) < ao. (42c)nG-o
Similarly, the complex cepstrum has the properties
x(O) = o n > 0 (43a)
O(0) = log x(O) (43b)
G
19-' --00
For rational z-transforms, X(z) has the form
m
B 11 (1-b kZ)X (z ) P O
H (I-dkz)k= 1
where the bk and dk are all less than one in magnitude, and the region of convergence is
25
Izi <rain jdI 1.k
In general, there may be poles and/or zeros on the unit circle. These cases willbe formally excluded from either class; however, we shall see (section 2. 7) that it is
possible to move such poles and zeros inside or outside the unit circle by exponentialweighting of the sequence.
If the input sequence is known to be minimum-phase, we can obtain significant sim-plifications in our results. We have already seen that the characteristic system D andits inverse can be realized through a recursion formula. We now wish to show that the
properties of minimum-phase sequences allow other simplifications in the computation
of the complex cepstrum.
Let us introduce some definitions. We define the even part of a sequence to be thesequence whose values are
Ev [WWI =x(n) + x(-n)
The sequence Ev [x(n)] is seen to have even symmetry; that is,
Ev [x(n)] = Ev [x(-n)].
Similarly, we define the odd part of a sequence as
Odd MI =x(n) - x(-n) (5Odd [x(n)] = 2 '(45)
which has odd symmetry; that is,
Odd [x(n)] = -Odd [x(-n)].
It can be shown that if x is a real sequence and
X(ej•) = Xr eJW4) + jXileJ4),
Od Xr (e is the transform of Ev [x(n)] and similarly. jXi(eJw) is the transform ofOdd WIxn]
Let us assume that x(n) = 0 for n < 0. In this case we can see from Eq. 44 that
x(n) = 2 Ev Ix(n)] n > 0
= Ev [x(n)] n = 0
=0 n<O.
That is, knowledge of the even part of a sequence that is zero for n < 0 is sufficient to
determine the entire sequence.
These properties represent a real part sufficiency theorem for z-transforms of
26
sequences that are zero for n < 0. For example, suppose we are given the real partX r(eJ W) of the transform of a sequence x. Since Xr(eJ) is the transform of the evenpart of the sequence, we can determine Ev [x(n)] from Xr(is)). If x(n) = 0 for n < 0, we
can determine the sequence x and therefore X(e ). Thus knowledge of Xr(eJ') is suf-
ficient to completely determine X(eJW).
Similar relations hold between the logarithm of the magnitude and the phase of a
z-transform, but under more restricted conditions. Rather than focus our attention on
relations between the magnitude and phase, let us consider the complex cepstrum first
and return to this question eventually. If x(n) = 0 for n < 0, we can apply the previous
results to the complex cepstrum to obtain
(n) = 2Ev [n)] n > 0 (46a)
= Ev [(n)] n = 0 (46b)
= 0 n < 0. (46c)
Since the real part of X(e W) is just
X r(e() = log IX(e3")I,
we see that
Ev rx(n) - log j IX(e')) ej3n d.
Thus, if (n) = 0 for n < 0, we only need to compute
Xr(ejo) = log IX(e3')j,
and we do not need to compute the phase.
If we wish to obtain the original sequence from log X(eJ3 )l, we can do so by first
computing Ev [r(n)], then ^ by using Eqs. 46, and then obtain the original sequence by
using the inverse characteristic system to compute x. Since the condition 2(n) = 0 for
n < 0 has been shown to be equivalent to the condition that x be minimum-phase, we
see that for minimum-phase sequences, the complex cepstrum can be computed from
00
M(e j) I x(n) e-jwn (4?)
n=O
Ev [x(n)]= -e log I X(eJ'O) 1 dw (48)
X'(n) =Ev [^X(n)] u(n), (49)
27
." .%ere
u(n) =2 n > 0
=1 n=O
=0 n<0.
Thus the characteristic system D for minimum-phase sequences can be realized as
ahown in Fig. 12, where "Fourier Transform" means X(e3 •).
Clearly, the discussion above also indicates a unique relation between log I X(eJ')l
FOURIER REAL FOURIER -TRANSFORM LOGARITHM 1X~n)X(.W) og X(e 1W) TRANSFORM Ev [x^(n)]
Su(n)
Fig. 12. A realization of the characteristic system D for minimumphase sequences.
and arg [X(eJ1)], In fact, the operations illustrated in Fig. 12 are equivalent to using
the Hulbert transform24 to obtain the proper phase curve for log I X(eJ) I when X(eJ()-
Ls minimum-phase.
As an example of the use of this result, let us consider the minimum-phase sequence
x(n) =an n > 0
=0 n<O.
The z-transform of this sequence is
X(z) = for jzj > jai,1 - az
and
log IX(el")j -I log (I+a 2 -2acosw).
Therefore the even part of the complex cepstrum is
Ev rx(n)]= -T-j log (1+a -Za cos ) ejwn dw
which can be written
Ev [(n)] = - log (l+az-2acosw) cos wn dk.
28
At this point, let us note two useful equations which can be found in Carslaw. 2 1
log (I+aZ-2acosw) dw 0 lal 4 1
=• loga 2 a~l (50)
and
log (1+a 2 -Zacosw) cos wn dw = -ar a n10Inaln l
a - J•-nl Jal (51. )
Inl
These equations hold for n an integer. We can use (50) and (51) to show that
Ev[ f(n)]= 0 n 0
InI al- Inj
Therefore we see that
nx (n) 2 Ev" ['x(n)] a n > 0n
-0 n <0.
In conclusion, we wish to cgL. attention to an interestinp representation of the input
sequence, and an interesting result for finite-length sequences. It is clear from the
properties of minimum-phase and maximum-phase sequences that every sequence x
may be expressed as
X = Xmin 'Vxmax,
where
Axmin (n) = •(n) n >, 0
A
Xmax(n) -- (n) n < 0.
1, or rational z-transforms, this is equivalent to
X(z) X min (z) X max(z),
where
Z9
m
k= I kzXmin(z) = PiI11 (1 -ck z-
k= I
11 (1l-b kZ) } ,
X ()k=Imax PO
II (l-dkZ)k=1
The results given here have particular significance for finite-length sequences.
Suppose that X(z) has the form
m mo
X(z) = A h~ (1-akz-') H I (-b kz),k= 1 k= 1
where the a, and bk are all less than one in magnitude. Clearly,
'mi~n(n) 0 0 •< n 4 m i
= 0 elsewhere
and
xmax(n) 0 -mi n •0
= 0 elsewhere.
From Eqs. 30 and 32, we obtain the relations
e•(O)xmin(n) =e n= 0
n-I
^ Xn) x(O) + ^k)x (- n >0k=O
mx (n) =I n = 0
0
k=n+ I
We see, therefore, that only m + m7i + ; values of the complex cepstrum arerequired to completely ,'eterrmine the mo0 + mi + I values of the sequence x. This
Th0
I __ _
result implies that even though x is of infinite duration, only a number of samples
of x equal to the length of the input sequence x is required to completely determine
the sequence x from the complex cepstrum.
2.7 EXPONENTIAL WEIGHTING OF SEQUENCES
Because of the special properties of minimum-phase sequences, it is of interest
to consider ways of obtaining minimum-phase sequences from nonminimum-phase
sequences. One way of doing this is to weight the nonminimum-phase sequence with
a decaying exponential. By this we mean multiplication of the values of a sequence by
na to obtain a new sequence whose values are
w(n) = anx(n).
There are two important points to consider. First, we shall consider the effect of expo-
nential weighting on a convolution, and then the effect OL the z-transform and its region
of convergence.Suppose that x(n) is given by
x(n) = Xl(k) x 2 (n-k).k= -co
For the exponentially weighted sequence, we obtain
00 00
w(n) an n x (k) x2(n-k)= a kxl(k) n-kx2 (n-k)k=-oo k=-oo
00
k=-ooWl(k) w2 (n-k).k= woo
Therefore exponential weighting of a convolution of two sequences x1 and x is seen to
be equivalent to the convolution of the exponentially weighted sequences w1 and w2 whose
values are
wI(n) = nxI(n)
w2 (n) = anw2(n).
The second interesting point is the effect of exponential weighting on the z-transform
of a sequence. The z-transform of the weighted sequence w is given by
00 -
W(Z)n= anx(n) z-n= Xa-l z).n= -oo ia
31
Therefore we see that if X(z) has a pole or zero at z = zo, then W(z) has a pole or zero
at azo. Thus if the region of convergence of X(z) is
R+ < IzI < R_,
then the region of convergence of W(z) is
aR+ < jzj < aR_.
If we have a sequence x for which x(n) = 0 for n < 0 but is nonminimum-phase, then
the sequence can be made minimum-phase by appropriate exponential weighting. We
simply need to multiply x(n) by an, where a is less than one and small enough to move
the pole or zero with greatest magnitude inside the unit circle,
We see, then, that exponential weighting may be very useful because convolutions
are preserved and it permits a more desirable pole-zero distribution. We should point
out, however, that if the required value of a is too small, we shall often be troubled
with rounding errors in carrying out such weighting on numbers stored in a computer.
A final point should be made. Exponential weighting clearly changes the complex
cepstrum. As the value of a approaches the reciprocal of the magnitude of the pole or
zero that is farthest from the origin, the complex capstrum becomes zero for nega-
tive n. If, on the other hand, a is close to 1, it will not significantly affect the complex
cepstrum, unless the z-transform of the input has poles or zeros on or close to the unit
circle. Since convolutions are preserved in the weighted sequence, the complex cep-
strum will always have the form
w (n) = wvI(n) + w 2 In).
In general, there is not a simple relationship between w1 (n) andxI(n); however, if the
poles and zeros of YX(z) are inside the unit circle, then clearly those of W (z) will also
be inside the unit circle if jaf < 1. Thus, in this special case, if
w 1 (n) = nx (n).
then
w (n) = a x (n).
Although there may not always be such a simple relationship between wI (n) and xI (n)
and W2 (n) and x2 (n), we dc still have a way of recovering xl(n) or x 2 (n) if we are given
wl(n) or w2 (n). This is so because
x1 (n) = anw I (n)
x (n) =anw(n).2 3
32
Owi
These relations are useful in practice, sincc mey allow us to work with exponentially
weighted sequences as inputs to the system D. If we desire the original sequences at
the output, we can simply unweight, using the relations above. In practice, this idea
is useful if we are dealing with finite-length sequences such that x(n) = 0 for n < 0 and
n > M, and if we can choose a so that a is not so small as to introduce excessiverounding error.•
2.8 MORE GENERAL RATIONAL z-TRANSFORMS
Let us assume that the z-transform of the input to the system D has the form
m. mr, 1 (1 r~ (I-b kZ)
XMz) = Azr k=( k=1 (52)Pi Po11 (1-Ckz-) II (1-dkz)k=1 k=1
In al of our previous results based on rational z-transforms, we assumed that A was
positive and real and r = 0. This was to insure that arg (X(z)] could be defined as
single-valued and continuous. Clearly, there are many interesting sequences that do
not have z-transforms of this form. For example, if we allow A to be positive or neg-
ative and r * 0, we can include most sequences of interest for computation. In fact,
finite-length sequences have z-transforms of the form of Eq. 52, with the ck and dk all
equal to zero. (Note that we have excluded zeros on the unit circle. These could be
included in our discussion if we were willing to consider discontinuous phase curves and
logarithmic infinities in the log magnitude. For simplicity, we shall take the point of
view that zeros on the unit circle have been shifted inside by exponential weighting.) Let
us now see how the results previously presented can be applied in this more general
situation.
When X(z) is actually the product of two or more z-transforms, we shall assume
that each term in the product is written in the form of (52). Thus the constant A will
be the product of the corresponding constants of the individual factors of X(z). For
example, if
X(z) = XI{z) . X2 (z),
then A
A =AAI A2 .A 2'
Clearly, A will be positive if AI and A2 are both of the same sign, and A will be nega-
tive if the signs of A1 and A2 differ. That is, by consideration of the sign of A it will
only be possible to determine the sign of A relative to the sign of A2 . In most situations,
33
the appropriate signs will be clear from consideration of the source of the signals or,
in many cases, the signs will not be important.
The constant term A contributes an integer multiple of v to the phase. Since wecan only determine whether A is positive or negative, we normally test to see if A is
positive or negative before computing the phase. If A is negative, we can change the
sign of X(eo j+) to effectively remove any contribution to the phase which is due to the
sign of A. Whether A was positive or negative can be remembered if this information
is of interest. The sign of A can be determined by noting that
m m
I[ (0-ak) [I (i-bk)
X(1) = X(ej 0) = A k=I k=l (53)pi POII (l-ck) II (l-dk)
k= I k= I
Since the ak, bk, ck and dk are all less than one in magnitude, all of the factors in (53)
are positive; therefore, the sign of A is the sign of X(1).
Let us now consider the effect of the factor zr in (52). Assuming that the phase is
computed as specified in section 2. 1, we can write formally
r M. mXz log + log A (54)
pi POIl I (-Ck z-1)I1 (l-dkZ)j
k=1 k=I
AThus, the complex cepstrum x consists of a component having all of the propertiee that
we have previously discussed and a component that is due to the term log [zr]. To see
how our results are modified by this term, let us consider the phase contribution for
z = e We are tempted to write
log [zr] = log [e r ejorJ = or + jwr.
If we recall, however, that the phase angle must be periodic in w (since it is the imag-
inary part of a z-transform), we see that arg [e(a+jw)r] must be defined as in Fig. 13.This factor then adds a nonanalytic component to the imaginary part of log [X(eq+jW)].
Formally, the contribution to the complex cepstrum of this type of term is
0(n) =e 2 r log fec+j] ejwn
Performing the indicated integration shows that
O(n) = r encos n0 (55a)
= r n =0. (55b)
34
r~ +j~)r
Fig. 13. Phase curve attributable to a factor zr when z e
This sequence is stable only if the contour of integration is the unit circle (a= 0). In
fact, the sequence, strictly speaking, only has a discrete Fourier transform
o0
O(eJ )= O(n) e'(Jcn,
since log z has no Laurent series expansion about z 0. This situation is analogous
to the continuous-time function (+t) ,which has no two-sided Laplace transform but
does have a Fourier transform.
Usually, we prefer to remove the linoar-phase component before computing the
complex cepstrum. This is easily done once the phase curve is computed, and
clearly its removal simply corresponds to a shift of r samples in the input sequence.
This value of r can be saved and used to shift the output of Dl, if this is appro-
priate. The parameter r is very much like the sign of A, in that if X(z) is the
product of X (z) and X2 (z), each having the form of Eq. 52, then r = r + r and
it will only be possible to determine r 1 and r 2 from consideration of the source of
the sequence x.
Thus, the complex cepstrum of a sequence whose z-transform is of the form of (52)
is normally obtained as follows. Choose the contour C to be the unit circle, and
find the contributions that are due to all of the factors except zr, using either the power
series expansion or the integral relations. We may then simply add to this the
component 0(n) given by Eq. 55. For example, we could write for sequences whose
z-transforms are of the form of (52), and choosing (r 0
35
"IT
Scoswn 1 X'(z) nx(n) -z dz n 0 (56)n T C X(z)
=log A n = 0. (57)
We note that the integral in (56) may be evaluated for n = 0 if we do not divide by n.
In fact, it is easily shown from the principle of the argument that the value of r can be
obtained from
r=- (z) .2wj X(z)
2.9 EXAMPLES OF COMPLEX CEPSTRA
We shall give several examples of the analytical determination of the complex cep-
strum. These examples, although simple, are somewhat typical of the kinds ofsequences that will be encountered in practice.
Example 1: Minimum-phase sequence
Let x(n) have the form
x(n) =0 n < 0
=an n ;%0,
where jaj <1.The z-transform of this sequence is
00
X(z) = anz-n = l for Izi > Ial.n=0 1 - az-n--0
The complex logarithm of X(z) is
X(z) = -log (0-az-l).
Since IaI < 1, we see that x is a minimum-phase sequence.
Let us choose the region of convergence of X(z) to include the unit circle so that this
can be the contour of integration in determining the complex cepstrum. In this case wecan obtain X(n) by three different methods.
(a) Integral Formula
Since we observe that arg [X(ejw)] is continuous everywhere for this example, k(n)
is determined by the equation
36
A = -zX' (z) n-I
x(n) Zfl X z) dz,C X(z)
where -zX'(z)/X(z) is given by
-zX' (z)
Xlz) I az--l
Thus we see that
(n) = i az- I zn- dI 2j a dz.-
I1 - az- I --jn z '--
This integral can be evaluated by using the residue theorem, to obtain
A nX(n) =-2"_ n1 > 0
= log (1) =0 n= 0
-0 n1<0
(b) Power Series Expansion for X(z)
A
Recall that X(z) is given by
Io aa nnX(z) = -log (0-az) = z -TZ
n= 1
where the power series expansion for -log (I-az-) is valid in the region JzI > jal. B,
definition, X(z) is also given by
ooXlz = •(n1) z-1
1n= -00
By comparison of the two power series, we see that
x (n)1 0 n 4 0
a'Aa- n >O0.n
(c) Recursive FormulaofA
Let us compute several values of x(n), uning the recursive formula. Since x(O) = 1,
1(O) = log xO) = 0.
37
I
iBy applying the recursion formula, we obtain
x(O)
A X(2) lA 2 2(2) x() ) x(1) = a' - a2
x(O)
A x(3) 3) a 3X(3M 4 ( 2(1)x(2)+2÷^(2)x(1)) = a' - (a +a =
We note that because the sequence is minimum-phase, the complex cepstrum is zero
for n < 0, and that our result agrees with that obtained by using the Hilbert trans-
form.
Example 2: Nonminimum-phase finite length sequence
Let x(n) be jiven by
x(n)= 0 for n<Oandn-M
=bn 04n<M.
The z-transform is given by
M-1
X(z) = z = I - bMz-Mn=O z
Let us assume that JbJ > 1. Then X(z) has M - I zeros located as shown in Fig. 14.
-b M M
Fig. 14. Zeros of Xlz) -- bz-
I - bz-I
Since the sequence is finite-length, the region of convergence of X'z) is the entire
z plane except for z - 0. If we write X(z) in the form of Eq. 52, we obtain
38
K
Xz=bM-1 -(M-I) (l-b-MzM
X(z) b z -1(I-b z)
The complex logarithm of this expression is
•(z) = log bM-I + log -(M-1) + log (l-b-MzM) - log (I-b-1 z).
If the region of convergence of X(z) is chosen to be the region I z I < fbj. then
A -Mn MnX^z bM -.(M-l) b-n n b MnX(z) = log bM + log + - n I n
n= I n= I
If we introduce the symbol 6(n)
6(n)= I n= 0
0 nJ•O,
we can write the inverse transform on the unit circle as
00 -1M-1 (1-M) cos wk Mk
(n)= ( bogb ) 6(n) + -k 6 (n-k)+ 6(n-kM)
k k=-oo
-1 k
S- •j --- 6(n-k).
k=-w
Therefore we observe that we obtain a stable sequence that is nonzero for both positive
and negative n. We can also see that if the sequence were shifted to the left by M - 1
samples, we would remove the term
00 (0-M) cos kk 6(n-k)Zk
k= -0o
from the complex cepstrum. That is, the sequence whose values are
s(n) = x(n+M-1)
will have the z-transform
NI-i (I-bN zMS(z) = b -
(i-b 7.)
Thus we can easily see that the complex cepstrum A(n) will be zero n > 0. This is so
39
because the shifted sequence is maximum-phase. The shifting of sequences to remove
linear phase components is quite important and will be discussed in detail in Section III.
Example 3: Repeated pulses
Suppose that the sequence x has values
x(ki) = s(n) + s(n-no) + ... + s(n-(P-1)n0 ),
where s is an aperiodic sequence like those of Examples 1 and 2. The sequence x can
also be expressed as
X = s U,
where u is a sequence whose values are given by
P-1
u(n) = I 6(n-kno).
k=O
The z-transform X(z) will have the form
X(z) = S(z) • UM)
where U(z) is
UM= -1 n 0 k =(1_zPno)
k=O ( no)
The region of convergence of U(z) is the entire z-plane except for z = 0. We note that
U(z) has zeros at equal angular spacing around the unit circle.
The complex logarithm of X(z) is
A% A AX(z) = log [S(z)] + log [U(z)] = S(z) + U(z).
Let us now choose a region of convergence for X(z). In this case it is not possible to
choose the region of convergence to contain the unit circle, since U(z) has all its zeros
there. This means that if S(z) is non minimum-phase, it will not be possible to obtain
a stable complex cepstrum. We can remove this difficulty by u.sing exponential weighting
of the sequence x. We know that if
x1 (n) = anx(n),
then
X (z) = X(a z) = S(a- z) . U(U- z) = S (z) , U (z),
where S (z) is the z-transform of the sequence whose values are
40
IAs1 (n) a ans(n),
and U (z) is the z-transform of the sequence
ui(n) = an u(n) =P- ank 6(n-kn)".k=O
Thus U (z) is
(1 no -rno
(I -aAz•)
Now, we see that the complex logarithm of X1 (z) is
i _Pnoz-Pno) lo 1_no -no).XI(z) = log [SI(z)] + log [UI(z)] = SI(z) + log ( [-Glog -a-anznI"
f Since the zeros of UI(z) now lie on a circle with radius Ia1, we see that the sequence uIcan be made minimum-phase by making Ij < i. In this case It would be possible to use
the unit circle as the contour of integration for obtaining x (n), which would be
o(o + nok @0 Pnok
xlI (n) = s-l(n) + -- E- 6(n-kn0 ) - -.- k
k= I k= I
Suppose that s is the sequence of Example 2, after shifting it left by M - I samples,
that is,
s(n) = bn+M- 1 M-I <n-5 0
= 0 elsewhere.
Then the sequence x would appear as in Fig. 1 5a, where P = 4 and M > no. The samples
of the discrete-time function have been connected by a smooth curve for convenience in
plotting. The weighted sequence xI is shown in Fig. I 5b, and the resulting complex cep-
strum is shown in Fig. 15c.
This simple example points out several things that are true in general:
1. The input xI was the convolution of a minimum-phase sequence with a maximum-
phase sequence. The resulting complex cepstrum consists of a part that is zero for
n < 0 because of the minimum-phase part of the input and a part that is zero for n > 0
because of the maximum phase part of the input. This is always true because any
sequence can be expressed as the convolution of a minimum-phase sequence and a
maximum-phase sequence (with possibly some time shift). In some cases either the
41
A
(b)
(c)
i p I I I I I
-3M -2M -M 0 n 2n 3no 4no 5n 6n 7n 810
Fig. 15. (a) Input sequence x. (b) Exponentially weighted sequence x1 (n) = anx(n).(c) Complex cepstrum of x .
minimum-phase part or the maximum-phase part msy be all that is of interest, while
in general this may not be true. When it is true, considerable simplification results
if we take advantage of the properties of minimum-phase sequences.
2. The input sequence uI had samr'es spaced at intervals of no, rather than 1. The
resulting complex cepstru.m has its samples at the safne spacing no. This can be shown
to be true in general if th,, spacing of the samples is uniform. It is also true approxi-
mately if the samples are unequally spaced, but it is difficult to obtain precise results
on this problem. Examples of this are given in Section V.
3, The input sequence sI was "pulselike"; that is, it had most of its samples con-
centrated in a small region relative to the total duration of u 1. The samples of s1 were
sp.,ced at urt intervals. The complex cepstrum of sI is seen to approach 0 -,j (ab)-r"/n.
so that it dies out at a fairly rapid rate, We have seen that it is true in general that the
complex cepstrum dies out at least as rapidly as I 1/nI for all n.
2. 10 LINEAR SYSTEM IN THE CANONIC REPRESENTATION
We have discussed in detail the an.--!:, .s and realization of the characteristic sys-
tem D (and therefore its inverse D ). We wish now to discuss the general type of lin-
ear system that has been found useful in separating convolved signals. We shall leave
specific examples to be covered in the sections on applications.
42
L!...
As a very general comment, we can say that the type of signals for which homomor-
phic deconvolution has thus far been found useful are those that are convolutions of sig-
nals whose complex cepstra, in some sense, do not overlap. An obvious example of this
is when one signal is minimum-phase and the other is maximum-phase. This was the
case in Example 3. The second basic situation is also indicated by Example 3. We
observe that the complex cepstrum of the "impulse train" uI has isolated samples
occurring with spacing no. Suppose that s8 is a sequence whose complex cepstrum dies
out rapidly so that sl(n) is small for n _< no (s, may not in general be maximum-phase
as in the example). If we have as the input
x I = sI 1 u l,.
Athen we see that the complex cepstra sI and 1̂ will be in a sense separated in "time" in
the complex cepstrum of x 1 .
Both of these situations suggest that the kind of linear system that we should use is
of the formI' A
(in) = 1(n n) . (58)
Such systems will be called frequency-invariant, in analogy with time-invariant linear
systems in which we multiply z-transforms and convolve time functions. For frequency
invariant linear systems we multiply time functions as in Eq. 58 and, therefore, we con-
volve frequency functions as in
=~ MleI L(ej(-) dg, (59)
where L(z) is the z -transform of the sequence whose valu--s are f (n).As a general comment, let us consider an interesting possibility that results if the
input is of the form
2 (n) ax I (n) + bx 2 (n).
The transform of this equation 1.
XMeL) aX1 (eJW) + bX.(eJW).
(See the Appendix for a discussion of scalar multiplication.) Suppose that we filter the
real and imaginary parts of X(e3W) with different linear systems Lr (e ) and Li(e
respectively., Thus Y(eJ ) would be of the form
Y(e3 ") =I- Xr(e ) Lr (e ()• ,' + 9-•) l L.le(-) d.1T -43 T
43
-- I
I I
OD)(n) In M I
L
(b)
Fig. 16. (a) Frequency-invariant system that is linear for real scalars.(b) General linear frequency-invariant lineai" system.
It can be shown that if and only if a and b are real numbers,
Y(e 3W) aY (e1) + bY (e )1 2
Thus for real scalars we may filter the magnitude and phase with separate filters.
Therefore we see that in general the linear system can be of the form
Y(n) = fr(n) Ev [2 (n)] + Ii(n) Odd [2 (n)], (60)
where
2 (n) = Ev [2 (n)] + Odd [2(n)],
and fr (n) and i(n) are the inverse transforms of Lr(e j) and L (e?'), respectively. The
operat-ns suggested by Eq. 60 are summarized in Fig. 16. Figure 16a illustrates the
general case and Fig. 16b, the case
r (n) = i(n) = (n).
44
III. COMPUTATIONAL CONSIDERATIONS IN
HOMOMORPHIC DECONVOLUTION
We have given a detailed analysis of the canonic form for homomorphic deconvolu-
tion. This analysis was carried out for discrete convolutions and it leaned heavily on
the z-transform both in the analysis and realization of the system. When we turn to the
actual computational realization (in the form of digital-computer programs), we find
that we must somtwhmit modify the results of Section II. The main reason for this is
that the z-transform (uaually its unit circle evaluation) is a function of a continuous
complex variable z (or w). Since digital computers deal with finite collections of num-bers rather than functions, it is clear that we must be content with only a finite number
of values of the z-transform. Thus we are led to the study of the sampled z-transform
and its properties.
The other major consideration is the calculation of the phase of the sampledz-transform. We shall apply the results of Section II to show what properties the sam-pled phase function must have and then we shall show an algorithm with which we can
compute tie proper phase.
Therefore our present major purpose is to show how the ideas of Section II can be
translated into programs for a digital computer. These programs will constitute our
realization of the canonic system of Fig. 4.
3. 1 SAMPLED z-TRANSFORM
As we can see, the use of the z-transform is very convenient in the analysis of
homomorphic deconvolution. If we wish, however, to use a computer to evaluate the
relations that we obtained in Section II, we must deal with only a finite number of values
of the z-transform. For the same reason, we shall be limited at the outset to finite-
lngthlength sequences.
Let us consider a sequence x whose z-transform has a region cf convergenceincluding the unit circle. (This will always be true for finite length sequences.) We can
therefore evaluate X(z) for z = eJ3 so that we obtain
00
X(e) =I x(n) ejwn. (61)n= - o-
Henceforth, (61), which is a function of the continuous variable W, will be referred to
as the Fourier transform (FT) of the sequence x. The inverse Fourier transform
(IFT) of X(ejW) is
1 X(d" du wn 1 2 W iix(n)X(e) =n dw = X(ej) ew dw. (62)
We note that the limits of integration in (62) can be any convenient interval of length 2w,
45
since both X(e3W) and edn are periodic with period 2w. For computation on a digitalcomputer, we must be content with only a finite collection of samples of Eq. 61. This
leads us to consider a sampling theorem for the transform X(ejw). Let us suppose thatwe have samples at exactly N values of w, that is, at N points around the unit circle
spaced at equal angular increments of 2w/N. Thus we obtain the sequence of samples
X N x(n) e for k = 0, 1 ... N-1. (63)
(We choose to use positive values of k for computational convenience.) Corresponding
to (62), we have
N-I L (J k) iL' kx(n)= N X e. (64)
k=O
To see how x(n) is related to x(n), let us substitute (63) in (64). The result is
x(n) N-I x(m) exp(-j 'km exp(j-•f kn).
k=O m=-oo
If we intercharnge the order of summation, we obtain
x(n) = x(m) exp 2W k(n-m (65)
M0 \ k=O
Let us define
N-I
d(n-m) = Y exp(j--wk(n-m)). (66)k=G
It can be shown that the sequence d is periodic with period N and that it can be
represented as
d(n-m) - (n-m+rN), (67)
where
6(n)-{ n#0
46
Therefore (65) becomes
No0o00
x~)x(m) d(n-m) = x(n+rN). (68)
m=-oo rM-00
Equation 68 shows that X(n) is periodic with period N, and that it is possible for x(n) tobe equal to x(n) for certain values of n if the original sequence is of finite length. Forexample, let us assnme that the sequence x has values x(n) such that x(n) = 0 for n < 0and n > M. It is clear from (68) that if N >M, then
x(n) = x(n) for 0 <n < N.
On the other hand, if N < M, we encounter aliasing in attempting to return to thesequence from the samples of X(eJ). This is shown in the simple example of Fig. 17.Figure 17a shows the original sequence x, wherein we can see that the length of thesequence is 5 samples (M=5). Using (66), we have plotted the sequence x for two dif-ferent values of N. In Fig. 17b we show the sequence x when we have sampled X(tjw)
0-00se pp e g .
e 0
0 00 0
0 06 0 0 0
Fig. 17. (a) Finite-length nequence of 5 sam~ples. (b) Periodic sequence cor-respndin tosampling the z-transforzn at 5 points on the unit circle.
(c) Periodic sequence corresponding to sampling the z-tran~sform at4 pontson he nitcircle. (Open circles indicate that two values
ovelapat 0.*4, *8.....This effect is called aliasing.)
47
five times (N=5). In the interval 0 < n < 5, x(n) = x(n), and the sequence x is periodic
with N = 5. In Fig. 7c x is shown for the case in which we have sampled X(ejw) only
four times. We note that x is periodic with period N = 4; however, two points of the
original sequence occur at the same value of n in x so that x(O) a x(n). This overlap
in the periodic sequence x is known as aliasing. Clearly, the way io avoid aliasing is
to insure that N > M; that is, we must sample X(eJ) at a high enough rate.
If we assume that N > M, we can write the following pair of relationships for finite
length sequences:
N-1 .2WX(k)= •_ x(n) e IT k 1 ...0 . N-1, (69)
n=O
= N-i 1 ~ k .2w,..N1.(01 J g kn
x(n) X(k) e n = 0, N-1. (70)
k=0
We note that both x and X are periodic with period N. That is,
x(n) = x(n+rN) for r = 0, *1, *2, ...
X(k)= X(k+rN) for r= 0,+I,+2.
Anot te way of expressing this fact is to say that in Eq. 69 and Eq. 70, all integers n,
k, and kn are to be interpreted modulo N. Equations 69 and '70 have been referred to
as a Discrete Fourier Transform (DFT) pair. 1 6
Let us now consider how (69) and (70) can be used in the realization of the charac-
teristic system D. We shall replace all z-transforms by Eq. 69 (the DFT) and all
inverse z-transforms by the inverse discrete Fourier transform (IDFT) of Eq. 70. Since
our interest is in convolutions of sequences, we must consider the effect of sampling
the z-transform when the input is a convolution. Let us assume that x and x2 are
finite-length sequences such that
xI(n)=0 for n<Oandn> M1 ,
x2 (n) = 0 for n<Oandn> M .
The convolution of x1 and x2 has values given by
n nx(n)= x I (n-r) x 2(r)= . xlI(r) x 2(n-r). (71)
r=0 r=O
We can see that the sequence x is also finite in length and that
48
g
x(n)= 0 for n<Oandn>M +M.
The FT of the sequenc'. x is X(edo) and is given by
X(eJ') = X (eij) • X 2(do).
Let us assume that we sample X(ed) at N points to obtain the samples
X/eJ" =Xeke N . Xe , k=0,1 ..... N-1. (72)
We have seen that if N is not large enough, aliasing will occur. To see the nature of
this aliasing for convolutions, let us note that (72) can be written in terms of the DFT.
X(k) = X (k) • X2(k) k = 0, 1 .... N-1, (73)
where k is taken modulo N. The IDFT of (73) can be shown to beSxj) x2(n)
0 0 0 0
MI M2
(0) (b)
0 x2(r)d
0
00 00
0:0
n-M n M2 r
0 000 000 01 0 0 0 0 0 0 0
0 00 0 0 0
-N ,N n-M n M2 N r
(e)
Fig. 18. (a) and (b) Aperiodic finit.e-length sequences. (c) Discrete
convolution of xI and x . (d) Periodic sequences for circu-
lar convolution when N < MI + M2 . (e) Periodic sequences
for circular convolutic, when N > I+ M
49
kIN-I N-I
r--i0 r-0
where r and n-r are both to be taken modulo N, andn= 0, 1, .... N-I. That is, x is
the result of a circular or periodic convolution as opposed to the original sequence x
which was the result of an aperiodic convolution. This point is illustrated by Fig. 18.
In Fig. 18a and 18b we show two finite length sequences of lengths M 1 and M2 , respec-
tively. Note that M < M2 . Figure 18c shows x (n-r) and x2 (r) plotted as a function
of r, as is required for convolution. The value of x(n) is obtained by adding the prod-
ucts xI(n-r) x 2 (r). From this figure, we can clearly see that x(n) will be zero for
n < 0 and n > M1 + M., since the sequences do not overlap for these values. Figure 18d
shows x, (n-r) and x2 (r) as would be obtained for N = M2 + 1. Although neither of the
sequences x1 or x2 differ from xI and x2 , respectively, in the interval 0 _< n < N, it
is clear that the circular convolution will not be equal to the aperiodic convolution for
this value of N. Figure 18e shows x,(n-r) and x2 (r) for N = M1 + M2 + 1. In this case,
it is clear that the periodic convolution x will equal the aperiodic convolution x in the
interval 0 - n < N. Thus, if we sample the z-transform at a high enough rate, the DFT
X(k) can represent an aperiodic convolution with no aliasing.
Let us now consider how we can compute the complex cepstrum using the DFT. We
recall that we have defined
X(ew) = log [X(edw)].A W
If we sample X(t{J) at N equally spaced values of w, we obtain
A( jLk~
XkeI= log k = 0, 1..., N-I.
The samples of X(eJw), of course, can be directly evaluated by using the DFT relation-
ship of (69). Therefore, we can write .
AX(k) = log [X(k)I k = 0,.. N-1, (75)
If we apply the IDFT, we obtain
N-I .2..E ki
2(n) log [X~k n 0, 1, N-1. (76)
k=0aAIn general, x is not a finite-length sequence so that we should expect some aliasing
A Ato occur in x. Equation 68 shows, in fact, that we must write x(n) as
00
x(n) = x(n+rN), (77)
so
i
where ý(n) is periodic with period N. The aliasing of the complex cepstrum may be a
problem in some cases and in others it may not. We recall that in general
I (n)l <A111 forall n *0,
so that it is possible, by making N large, to eliminate most of the aliasing of the corn-
plex cepstrum. By this we mean that if x(n) = 0 for n < 0 and n > M, then by choosing
N >> M and defining
x(n)= x(n) 0 n<M I=M M4n<N,
there will not be as much overlap between periods of • as if we had chosen N = M + 1,
as it is clearly possible to dc. Therefore the aliasing of ý is minimized by choosing
N as large as possible consistent with the computer storage and computation time con-
straints and then padding the M + I samples of x with zeros.
The numerical operations of computing the complex cepstrum are summarized in
the following equations.
N-i 2WX(k) -- I x (n) e k -k0, 1 ... N-1, (78a)
n=0
X(k) = log [X(k)J k 0, N-1, (78b)
N-I ZWj-k
N X(k) e n = 0, I. N-I, (78c)
k= 0
We recognize that the complex cepstrum that we compute by using these equations will
differ because of aliasing from the theoretical complex cepstrum obtained through the
z-transform.
3.2 FAST FOURIER TRANSFORM
In 1965, Cooley and Tukey! 4 disclosed an algorithm for high-speed calculation of
the DFT. Since that time, there has been tremendous interest in the application of thisalgorithm in many diverse areas. One of the field.- in which it has been used with great
success is that of digital filtering or waveform processing. 1 6 ' 17 In fact, the availabil-
ity of this algorithm allows us to consider realizing thid scheme for deconvolution.
To see the nature of this new method of evaluation of Fourier transforms, let us
recall the DFT pair
51
N-1 .2w" kX(k) = x(n) eN k 0, 1..., N-1, -?a,
n=0
N-I .•k
!(n) - X(k)~ e n =0, .... N-1. (79b)-N
The fast Fourier transform (FFT) is just Eqs. 79a or 79b done in a high-speed way. We
note that to evaluate Eqs. 79 in a straightforward manner involves N2 complex multi-
plications and additions. Cooley and Tukey showed that if N = 2 m, it is possible to
evaluate either (79a) or (79b) by using m iterations of a process involving N complex
multiplications and additions. Therefore the total number of operations is N log2 N,rather than N2. Clearly, if N is only moderately large, the amount of computation (and
thus computing time) is considerably reduced by using the FFT algorithm.S... .. 14-17Since the publications on the subject of the FFT have mushroomed in the
several years since the appearance of the original Cooley-T,,key paper, ,/e shall not
discuiss the algorithm and its programming. We shall, however, be interested in the
properties of the FFT (or DFT), inanj of which are slightly different from corre-
sponding ones for the z-transformr or Fourier transform of a sequence. This difference
is usually a result of the periodicity of both X(k) and x(n). We have already seen an
example of this in the case of convolution. A table of the useful relations and symme-
tries is given by Gentleman and Sande. 1 5
3.3 PROPERTIES OF THE SAMPLED-PHASE CURVES
We shall consider the problem 3f computing the samples of arg [X(eJw)] from the
samples of X(eJw). We have shown that arg [X(eJW)] must be continuous for -a < w < it
and that it must be odd and periodic with period 2w. We recall that for rational
z-transforms, arg [XieJW)] is in general discontinuous at w = *w, *3, ... because of
the linear phase component. These conditions imply a similar set of conditions on the
samples of arg [X(e' )], that is, on arg LX(k)], f.r k = 0, 1, 2 .... N-1. These condi-tions are given below under the assumption thai we have sampled X(eJw) at a s,'*fi-
ciently high rate so that the conmplex cepstrum will not be severely aliased. These
ronditions are as follows.
CI. The inequality
Jarg[X(k)]-arg[X(k+I)J1 <,E
must hold forO0 < k< N/2.- I and for N/Z + J k< N-I, wh, re e is a toler-
ance depending on N (that is, the rate rf sarnplh.g of X(c")).
5z
C2. arg [X(k)] is an odd function of k; that is,
arg (X(k)] = .arg [X(N-k)J
for k = 0, 1. ... , N-l, with k and N-k taken modulo N.
C3. arg [X(k)] is periodic with period N; that is,
arg [X(k)] = arg [Xl(k+rN)] r = 0,*1,*2,...,
where k = 0, 1, ... N-I, and k and k+rN are taken modulo N.
To see what these conditions mean, let us consider the type of phase curves to be
expected for finite-length sequences. Consider a sequence x whose values x(n) satisfy
x(n)= 0 for n< Oandn>M.
The corresponding z-transform is
M M . m
X(z)= x(n) z Az ()-brZ)
n=O r=0 r=l
where the ar and br are all Jess than one in magnitude, and M mi + mno. Equation 80
places in evidence the fact that in general there will be m. zerois inside the unit circle
and m zeros outside the unit circle. Since arg [X(e3 )] is thu sum of the angles of each
factor in Eq. 80, consideration of Figs., 8, 9, and 13 shows that arg [X(eJw)] has the
properties
arg [X(eJO) 0 for w = 0, 2n,4w,.... (81 a)
rg [X(e)] = -mor for w = IT-y, 3•-Y.... (81b)
arg [X(eJA)] = m0w for w = w+V, 3r+y...., (81c)
where y is an arbitrarily small positive number. In obtaining these equations, we have
assumed that A is positive.
Let us now consider the corresponding sampled Fotirier transforra and its sampled.2w
phase. We obtain X{k) from (801, simply by replacing z by Wk, where W = e . There-
fore we obtaii-
_kmo mi m
X(k)= AW H (I-arW-k) IIn -brW (82)r- I rl rI
or k = 0, 1 .. N -1. We can determine the sign of A by looking at
f m m0
X(O) = A n (I-ar) II (1-br)~ r= I r= r
since the sign of A is the same as the sign of X(0). If A is negative, we normally change
the sign of X(k) before computing arg [X(k)I, so that we remove any constant phase
53
component caused by A. Henceforth we shall assume that A is positive.
The properties corresponding to Eqs. 81 are easilýy ýtu tu be
arg [X(k)] = 0 for k= 0,-2,N, (83a)
arg [Xk)] = -moIT for k 1, '2 1 . (83b)
arg [X(k)] +mo7 for k=-+ ON +N .. , (83c)0 2 12+1
We note that arg [Xlk)] is given the value 0 at k = N/2; 3N/2, ... so as to satisfy the
requirement of odd symmetry. These properties are exhibited by Fig. 19a where we
have shown only one half cycle of arg [X(k)].
org EX(k)]
N/2 k
-31I
ANG (Xlk)j Fig. 19.lWi
(a) Samples of arg [X(eJ)I. (b) Principal
N/2 k value of arg[X(eJw")]. (c) Correction. sequence for obtaining P.rg from ARG.
"(b)
CON (k)
N/7
-2w .4.4..-
-4v
(C)
These properties of the sampled phase curve wiln be used to discuss an algorithm
for compu.ing erg [X(k)] from X(k) which in turn is obtained by using the FFT algorithm.
3.4 AN ALGORITHM FOR COMPUTING arg [X(k)I
The properties of arg [X(k)] were given in section 3. 3. These properties must be
satisfied by any phase curve that we compute if we want to insure that
54
where x is a finite length sequence of the form
x = xI @ x2 .
Asampled phase curve is shuwn in Fig. 19a. In Fig. 19b, we show the principal
value of the curve in Fig. 19a. Let us recall that we can cbtain X(k) from
N-I
X(k) = x(n)W-kn =X (k) 4 J:i~k}, (84)
n=0
where N is a power of 2 that is greater than the length of the sequence x. Thus we are
given Xr (k) and Xi(k), and we must compute arg [X(-.k) so as to satisfy the properties
given in section 3.3. This is not as simple as it may appear at first glance. The prin-
cipal value may be easily obtained by using stardard rouiines based on a polynomial
approximation. Let us assume that we have computed
-r< ARG [X(k)] 4<T (85)
for k = 0, 1, .... N-I. Although ARG [X(k)] does not satisfy our requirements, it can be
used as basis for computing the correct phase curve. To see this, consider Fig. 19b.
We note that arg [X(k)] can be expressed as the sum
arg [X(k)] = ARG [X(k)] + COR (k), (86)
where COR (k) is shown in Fig. 19c for that example. In general, it is clear that
COR (k) = 27rq,
where q is a positive or negative integer that depends on k so that the properties of
section 3.3 are satisfied for all k.
Our discussion suggests that we can compute arg [X(k)] from ARG [X(k)] by com-
puting the correction sequence COR (k) and then adding it to ARG [X(k)]. This can be
done if the sampling rate is high enough. Then ARG [X(k)] contains all information
required to compute COR (k). In order to see this, it is convenient to make several
definitions. We say that ARG [X(k)] has a positive jump of 2,r (PJ of 2n) at k if
ARG [X(k+l)) - ARG [X(k)] > ZiT - E1 , (87a)
where Cl is a positive number whose value depends to some extent on the rate at which
we sample the phase. Similarly, we say that ARG [X(k)] has a negative jump of 2w at k
(NJ of 27r) if
55
ARG [X(k+l )] - ARG [X(k)] < -(21-Ec). (87b)
By carefully considering the example of Fig. 19, we see that COR (k) can be com-
puted by using the following algorithm:
COR (k+l) = COR (k) - 2ir for a PJ of 21 at k (88a)
COR (k+l) = COR (k) + 2i for a NJ of 2Z at k (88b)
COR (k+) = COR (k) otherwise, (88c)
where COR (0) = 0 and k = 0, 1, .... N-1. Therefore we see that we can compute
arg [X(k)] in the following steps: Compute ARG [X(k)I for k = 0, 1, ... , N-I, using a
suitable routine based on an inverse tangent approximation; use Eqs. 87 and 88 to com-
pute COR (k) fork= 0, 1, ... , N-I; and add COR (k) to ARG [X(k)] fork= 0, 1, ... , N-I
to obtain arg [X(k)]. (We note that this step can be done, as COR (k) is computed so that
extra storage for COR (k) is not required.)
When we are dealing with input sequenccs with many samples, we quite often find
that there may be several hundred zeros of X(z) outside the unit circle. Since
argx-moIT,
where m is the number of zero- outside the unR circle, we often find that the linear
phase component is so large that it dominates the rest of the phase components. Let
us call the sampled linear phase component ®(k) . Therefore we find that in order to
have properties of section 3.4, we require
(a(k) = - -- mok 0 N<k < (89a)N o
_N
= 0 k -N (89b)
Zi Nm(k-N) < k < N. (89c)
It can be shown that the contribution to x(n), which is due to jiQ(k), is
2(n) =m cosf• n n 1, 2,... N-1 (90)N 0tan rn
(90)=0 n=0.
These sequences are shown in Fig. 20. Just as @(k) dominates the phase when m0 is
large, we find that O(n) dominates the complex cepstrum and obscures much of the
interesting information in •.
56
(t b)
I
N-k
/I
~~00(b) IDF of (a)
I I
I tI
I I
, (b)
-kmFig. 20. (a) Linear phase component attributable to a factor W o.
(b) IDFT of (a).
It is possible to remove this component before computing the complex cepstrum.
Clearly, we could simply subtract the sequence in Fig. 20a from arg [X(k)] before com-
puting the complex cepstrum. This car. be done, since we note that
m. 0- = -XI - 0 (19)
if we have sampled the phase curve at a high enough rate. Thus we can compute the
right. hand side of (91) and round off the result to the nearest integer to obtain mi.
Removing the linear phase component in arg [X(k) is equivalent to removing the-km km
factor W in (801 by riultiplying by W o. This in turn is equivalent to rotating
(since all integers are taken modulo N) the input sequence x(n) to the left by m° posi-
tions. This fact may be used at the output of the inverse characteristic system to
reposition the output sequence when ýhis is necessary.
As an example, let us consider the phase curves of Fig. 19. We see that we can
compute Figs. 19b and 19c by using the methods discussed previously. If we add these
two, we obtain Fig. 19a which contains the undesirable linear component. On the otner
hand, if we first note that
57
I.A
ARG[X - Q] + COR !-"
m 0 - (92)11
then we could subtract 8(k) from COR (k) before adding it to ARG [X(k)I. This is
shown in Fig. 21 wherein we have repeated Fig. 19h as Fig. 2Ia. Figure 21a shows the
resulting correction for this example, and Fig. 21 c shows the sum of Fig. ZIa and 21b.
The result is the phase curve of the rotated sequence whcse values are .(nI+m 0 ), where
n + m is taken modulo N.
AMG rX(k)]
-~N/2 k
OW
(0)
COR (k) - etk)
2w
IF - F i g . 2 1 .4€ ....... I m N,2 k (a) Typical principal value curve. (b) Cor-rection curve for obtaining arg [X(k)] and
""v .at the same time removing the linear phasek (b) component. (c) Result of adding (a) to (b).
2v
Ic)
Let us summarize the results of the previous discussion. Our procedure in com-
puting the phase is as follows.
(AI) Compute the principal value ARG [Xl(k)j from the DFT X(k) for k = 0, 1,...
N- 1.
(A2) Compute the step correction function COR (k) for k = 0, 1, .... N-11 usingEq s. 8 8.
(A3) Determine how far to the left the sequence should be rotated by using Eq. 92.
(A4) Subtract ({k) as defined in Eqs. 89 from COR (k) for k = 0, 1, .... , N-I.
(AS) Add the result of (A4) to ARG [X(k)] for k = 0, 1 .... N-I to obtain the phase
for the rotated sequence.
58
[I
If one has enough computer storage for an extra table of N values, we recommend
that each step be done for all vallus of k before proceeding to the nex. step. This ia
because of the differences in the s;izes of the values of the three s-eqences that we are
"adding or subtracting. The maximum value of ARG [X(k)] is clearly w. The maximum
magnitude of COR (k) and of 8(k) is moow; however, the maximum magnitude of the dif-
ference of the two is generally much less than mo0.
If storage is not available for an extra table, we can accomplish the same result by
essentially doing (A2) twice. We can see from Fig. 19 that the value COR (N-I) is '
equpl to
COR (l 1 ) = 21T(#NJ-# PJ),
where #PJ is the number of positive jumps, and #NJ is the numnber of negative jumps in
the interval 0 < k < N/2. Thus we can compute m from Eq. 92 and form COR (k) - 0(k)
as it is added to ARG [X(k)] and then store the result in the same location.
3.5 OTHER COMPUTATIONAL CONSIDERATIONS
We shall discuss some simple techniques that can be used to minimize the required
computation time for realizing the characteristic system and its inverse.15
Gentleman and Sande summarize some of the useful properties of the DFT. Many
authors refer to the symmetries inherent in the DFT relationships. For example, we
find that if x(n) is real, then the real and imaginary parts of the DFT
X(k) Xr (k) + jXi(k),
j have the properties
X r (k) X r(N-k) (93a)
Xi(k) = -Xi(N-k) (93b) t
for k = 0, 1. N-I and k and N-k taken modulo N. Thus we say that the real part of
X(k) is an even function of k, and imaginary part tif X(k) is an odd function of k.
Let us consider a sequence whose values are purely imaginary such as
x(n) = jq(n) for n =0,1 .... N-i,
where the q(n) are real numbers. The resulting DFT would be
X(k) = jQ(k) -Qi(k) + jQr(k),
where
Q(k) = Q (k) + jQ.(k)
59
is the DFT of the real sequence q whose values are q(n). Therefore, if
x(n) + p(n) + jq(n),
where p and q are both finite length sequences, then we see that
Xr(k) = r(k) - Qi(k)
Xi(k) - Pi(k) + Q (k).
Because of the symmetry properties, we see that the following relations are true.
P(k)- P r(k) + jPi(k) = Ev [Xr(k)] + j Odd [Xi(k)] (94a)
2(k) = Qr (k) + jQi(k) -- Ev [Xi(k)j - j Odd [Xr (k)], (94b)
where, for example,
X r (k) + X r(N-k)
Ev [r(k)] = k = 0, 1 ... N-1 (95a)
X r (k) - X (N-k)Odd r(k) = Z k = 0, 1, ... N-I. (95b)
From these relationships, several comments are in order.
Comment 1: For real sequences, if we are given X r(k) for k = 0, 1, .... N/Z, and Xi(k)
for k = 1, 2, ... , N/2-1, then we can use Eqs. 93 to determine Xr(k) and Xi(k) for "l1 k.
This fact can be used tU conberve memory when we must store several complex tra's-
forms for some reason.
Comment 2: We can obtain the DFT of two sequences p and q by using only one evalu-
ation v-f the DFT relatiunships by evaluation of
N-I
X(k) = (p(n)+jq(n)) W = P(k) + jQ(k).
n=O
We have seen that Eqs, 94 and 95 can be used to recover P(k) and Q(k) from X(k). Com-
ment I can be applied here to allow us to store the transforms P(k) and Q(k) in 2N loca-
tions, rather than in the 4N locations required to store all values of these transforms.
Comment 3: In computing I x(k)12 , 1 log [ X(k)12 , ARG [X(k)], COR (k), and arg [X(k)],
we recognize that each of these are even or odd so that we only need compute N/2 appro-
priate values of the sequences, and then we can find the rest of the values by symmetry.
Thus we can save almost one half of the time in performing these operations on X(k).
60
3.6 COMPUTATION TIME REQUIREMENTS
We have shown the numerical techniques that must be used to carry out the trans-
formations D and D71. A major consideration is the amunt of time required for these
operations. We shall provide rough guidelines for estimating the computation time.
i FOURIER ~ FOURIERTRANSFORM FUIR,
Fig. 22. Computational realizati-n of the characteristic system.
The operations required for the transformations D and D-I are summarized in
tables of real numbers. The length of these tables is N, which is a power of two. As
we can see, these numbers are combined together in various ways to obtain the oper-
ations of FFT, log, ARG, etc. These numerical operations consist of additions, sub-
tractions, multiplications, and divisions, coupled together by logical operations and
indexing through the tables. Most of the total computation time is due to t.,e arithmetic
operations.
7_ -, CONVERi Y "--- t"•
ilFAST lYl FROM Y INVERSE /
FOURIER POLAR FORM F.A'J
TRANSFORMarg (Y] RECTAN.GULAR y, TRAN.;FORM1
~ ~~FORM ~
Fig. 23. Computational realization of tCe inverse characteri,3tic system.
In realizing the transformations D or D-, we require 2 FFT's. We have stated that
the number of complex additions and multiplications is equal to N log2 N. The remaining
operations; log, ARG, arg, exp, etc. all require a number of additions and multiplica..
tions proportional to N. Thus it is reasonable to write
TD ZTFFTN log2 N + TADN 96)
and
TDI ZTFFTN log, N + T A N, (97)
61
w..here
TD = time required to compute the complex cepstrum
T 1 = the time required to compute the output
TFFTN log2 N = the time required to compute the F 7T
TADN = the time required to compute the log magnitude and the phase cu,'ve
T 1 N = the time required to exponentiate the transform of y.AAD
For one realization on the TX-2 computer, we obtained the values
TFFT = 60 I'Lec
TAD:: 9 msec
T AD = 6 msec.
Thus, for extanple, for N = 4096, we obtain
TD =40 sec
T = 30 se¢D-
We should point out that Eqs. 96 and 97 would be divided by 2 if we exploit all of the
symmetry properties.
3.7 MINIMUM-PHASE COMPUTATIONS
We have discussed vertain simplifications that occur when the input sequence is
minimum-phase. As we have seen, for minimum-phase finite-length sequences, all
zeros of X(z) are inside the unit circle, and we have as many poles at zero as we have
-nonzero zeros. We have shown that, in this case, a recursive formula can be derivcd
for x(n) and we can also compute 'xn) from log I X(eJ")I alone without computing the
phase. We shall now consider the actual ccrnmutationai realh;,ation of this second method
and compare it with the recursive method.
We have seen that the samples of log I X(eJ)I may be coml.uted by first calculating
N-1
X(k)~ X,(k) + jXi(k) = x(n) W-kn
n=O
for k = 0, 1, .... N-I. !We use the FFT algorithm to evaluate these eq Ations.) We may
use a polynomiai approjximation to compute
(k) log I Xk)I log X[r(k) + X (k)
for k = 0, 1, ... N-1, as we Pormally d-, even if the phase is to be used. Because of
62
the periodicity of x and A, the even part of A is
A. Ax(n) + X(N-n)
E (n)]- ^
where n and N-n are taken modulo N. Since X (k) is an even function, its IDFT isEv [!(n)] so that
N-1Ev [^!(n)] l og Xr(k) + 1 k (99)
k= 0
for n = 0, 1 .... N-I. We note that •(n) = 0 for n < 0 for minimum-phase sequences.
We have also seen that if X(n) = 0 for n >_ N, then no aliasing will occur in 2 (n), that is,
A A!(n) = x(n) foi, n = 0, 1, ... , N-1.
We note that iff(n) 0 for n >_ N/2 and ii<0, it is clearly true that
A A!(n) =x(n) for n = 0,1...,N-1.
Furthermore, we see from (98) that in this case we can obtain A(n) from Ev j(n)]using the relation
A [Ax(n) = Ev [x(n)] u(n), (100)
where
u(n) I n = 0
=2 0<n<-n2 <
=0 N -<NN<N. (101)
AAif x(n) 0 for n>_ N12, asis ualythe cae e-nstill us q.99 and 100 to
compute an aliased approximation to x(n). Clearly, the more rapidly X(n) approaches
t zero, for large positive anJ negative vp'.es of n, the better will be the approximationS'for a given value of N. Alternatively, the larger we make N, the better will be the
approximation tc n In the case of minimum-phase sequences, if we exponentially
weight the input so that
w(n) = anx(n),
then the complex cepstrum is also exponentially weighted by the same exponential. That
is, if Ial < I,
) = a (n).
63
Therefore exponential weighting can be used tc help reduce the aliasing of the complex
cepstrum.
In summary, we can compute an aliased complex cepstrum when x is minimum-
phase by using the following equations which are the counterparts of Eqs. 60-62.
N-
X(k) x(n) W-kn k = 0, 1,... N-I (10Za)n=0 N-
! N-1
Evrf(n' j log X(k) I W n = 0, 1,... N-1, (102b)
k=0
!(n) = Ev [-(n). u(n). (J02c)
.2w
We have defined u(n) in (101) and we recall that W = e
"As an alternative, of course, we have Eqs. 44. These equations are repeated below
for the case x(n) = 0 for n < 0 and n > M.
x(n) = log x(O) n = 0
x(n) n k x(n-k)=--- L n (k) O--< n_< M
k=0 x(O)
xln •(k) M<n
x(n) x(n-k)--
M I nn.0x(0) k=n-M
It is clear that the recursive algorithm has the advantage ove.* Eqs. 102, in that noaliasing is introduced. Furthermore, to compute M samples of ý(n) we require only
2M data storage locations. The recursive algorithm, however, suffers greatly when
compared with Eqs. 102 on the basis of computation time. If we e:xploit all the sym-
metry properties in carrying out Eqs. I0V as discussed in section 3.6, we find that
for values of M greater than 64, the evaloation of Eqs. 102 for N = 2M is faster than
evaluating M points by using the recursive algorithm. This is an estimate based on a
third-order approximation to the logarithm and the assumption that a multiplication takes
twice as long as addition. In cases irn which a higher order approximation to the loga-
rithm is required, or a multiplication takes longer than twice the time for an addition,
this crossover point will be higher, but not significantly higher. Therefore for minimum-
phase sequences we find that even though we have obtained a recursive algorithm in
which aliasing is not a problem it is usually preferable to use Eqs. 102 with N such that
aliasing is not significant. The recursive algorithm is still useful conceptually, and it
64
may find application when only a small number of values of ý(n) is required or when
aliasing cannot be tolerated.
3.8 SA'APLING OF CONTINUOUS-TIME SIGNALS
It can be shown that in the case of continuous-time signals whose Fourier transforms
are bandlimited, the samples of a continuous time convolution are equal to the discrete
convolution of the samples of the individual signals. That is, if time and frequency are
normalized so that the Nyquist frequency is 1 Hz, then
00
xln) 0 XlIT) xln--T) dT x I (k) x2 (n-k). (103)
k= oo
In our calculations, we have assumed that all sequences are of finite length. This
assumption would imply that the corresponding continuous time function is time-limited.
It is well known that a time-limited signal cannot have a frequency-limited Fourier
transform. In practice, however, the Fourier transform approaches zero quite rapidly
in most cases, even when the signal is time-limited. Therefore we shall use the approx-
imation that beth the continuous time signal and its transform are zero outside of some
finite interval.
Let us consider a continuous-time function x(t) whose Fourier transform X (W) is
such that
X (W= 0 L >
The Fourier transform of the sequence of samples of x(t) is
X(e j W) = Xc(W) for -ir < w <
and X(ej ) is periodic with period equal to 2n.
Such an example is shown in Fig. 24. There we see that if we sample exactly at the
Nyquist rate, we encounter no aliasing of X(eJW), and I X(eJw) is nonzero for all w. This
means that log I x(eJ•') will be finite for all w. This is shown in Fig. 25a. On the other
hand, if we sample at a rate higher than the Nyquist rate as in Fig. 24c, we see that
X(eJW) will be zero over a finite interval, so that log I X~eJw) will be undefined in that
interval. This is shown in Fig. 25b.
The previous example is idealized; however, the essential points remain true in
practice. If we are dealing with a finite-length sequence that is the result of sampling
some continuous-time signal, we normally would try to sample at a rate that is as high
as possible so as to avoid aliasing in the Fourier transform. Since the high-frequency
content of many signals (speech or seismic signals) is quite low, we sha.l normally find
that if the sampling rate is high, there is a significant interval of frequenc'y over which
65
I X (W) I
(a)
-3w -2w -w N 2w 3w
(b)
IX(eji)l
-W -it it W .
w WS
(0)
Fig. 24. (a) Magnitude of the Fourier transform of a bandlimitedcontinuous-time function x(t). (b) Transform of thesamples of x(t) for sampling just at the Nyquist rate.(c) Transform of the samples of x(t) for sampling at arate higher than the Nyquist rate.
66
log
-~~I ()IX(.iU)I
(b)
Fig. 25. (a) Log magnitaide fbr Fig. 24b.(b) Log magnitt..oe tor Fig. 24c.
X(ejw) is very small. This means that if we compute the sampled transform
N-1X(k) = Xr(k) + jXi(k) ~xn) W-kn k = 0, N-1,
n=0
then both X (k) and X.(k) will normally be quite small for values of k around N/2. Since
we must compute h,)th log IX(k)I and ARG [X(k)], using polynomial approximations
involving divisions, -me shall normally encounter severe accuracy problems when both
X r(k) and Xi(k) are very small. This is especially true in using fixed-point machines
in which we must effectively keep the scale factor of Xr(k) and Xi(k) the same for all
values of k. Similarly, the scale factor must be the same for log I X(k)j and arg [X(k)]
for all values of k. Therefore it is clear that in sampling a continuous-time signal so
as to apply discrete homomorpnic deconvolution, we must effectively sample as in
Fig. 24b, rather than as in Fig. 24c. In most cases we can usually lowpass filter the
signal before sampling. Therefore if we precede the sampler with a very sharp cutoff
filter, and then sample at a little more than twice the nominal cutoff frequency of the 4filter, we shall obtain a sequence in which aliasing is not excessive in the Fourier
transform, while at the same time we shall make it possible to accurately compute
log IX(k)I and arg [X(k)I for all values of k.
67
---------
IV. ANALYSIS OF SPEECH WAVEFORMS
We have shown that it is possible to transform a convolution of two or more signals
into a sum of related signals. We have seen that in some cases these signals are sepa-
rated in "time,, so that a frequency-invariant linear system can be used to advantage in
separating the signals from one another.
We shall now discuss a simple model for the production of speech waveforms. This
model results in a representation of certain segments of speech waveforms as a con-
volution of an aperiodic pulse with a quasi-periodic impulse train. Thus speech wave-
forms are examples of the class of signals to which this technique of deconvolution is
particularly applicable.
Our purpose in discussing speech is twofold. First, we shall see that speech wave-
forms serve as very interesting examples of the tec1hniques presented in this report. Sec-
ond, Section V will focus on the problem of echo removal and detection. As a particular
example of the application of the results of Section V, we shall discuss the removal and
detection of echoes in speech signals. Thus we shall also obtain an understanding of the
characteristics of the complex cepstrum of speech which will be applied in another con-
text in Section V.
The application of homomorphic deconvolution to speech analysis by Oppenheim 19 ' 20
has paralleled our application to echo removal. This section will attempt to give a brief
introduction to this field of application. The reader who is interested in this is directed
to Oppenheim and Schafer19 and Oppenheim20 for more detail. We shall first discuss
a model !or the speech waveform and then consider the complex cepstrum of speech. We
shall give some examples illustrating the applicability ol our techniques to the recovery
of the separate components of the speech waveform.
4.1 SPEECH PRODUCTION AND THE SPEECH WAVEFORM
The speech signal is an acoustic disturbance that is generated by air escaping from
the lungs of the speaker. The mouth and throat form an acoustic resonator called the
vocal tract which is excited by the air that is supplied by the lungs. We may think of
the lungs as a source of a steady flow of air which is converted into a varying flow either
by the vocal cords or by constrictions of the vocal tract. In the first case, the vocal
cords may convert the steady flow of air into a series of quasi-periodic pulses by rapidly
opening and closing the passage to the lungs. Sounds generated in this way are called
voiced speech sounds. Examples are the vowel sounds. The other principal class of
speech sounds is termed unvoiced and these are generated by creating constrictions in
the vocal tract which cause turbulence to occur at these points. Many of the consonants
are generated by using this type of "noise" signal as excitation for the vocal tract.
In Fig. 26a, we see a schematic representation of voiced speech production. 18 Thelungs supply a steady flow of air that is modulated by the vocal cords to give the excita-
tion function e(t) as shown in Fig. 26b. The vocal tract is modeled by a cascade of linear
68
resonato•.s whose combined impulse response v(t) is typically like Fig. 26c. The reso-
nant fr,=quencies of the resonators are called the formant frequencies. The speech
LUNGS VOCAL VOCALSTEADY CORDS 00, TRACT 90)FLOW Ca)to)
0(t) = p(t)0 g(t)
I _zt
Fig. 26. (a) Model of voiced-speech pro-(b) duction. (b) Excitation function
generated by vocal cords. (c) Im-VWt) pulse i" e s p o n s e of vocal tract.
(d) Resulting speech waveform.__t
(C)
I SO
(d)
signal is the response of the vocal tract to the quasi-periodic excitation e(t), and
therefore is given by
s(t) v() e(t-T) dT.
Clearly, the resulting speech signal is also quasi-periodic. Since e(t) is quasi-periodic,
we could further represent e(t) as the convolution of a basic pulse shape g(t) (called the
glottal pulse) with an impulse train p(t) such that
e(t) = g(t) @ p(t).
Ther-fore we obtain
s(t) = v(t) 0 g(t) 0 p(t).
In the case of unvoiced sounds, the vocal tract is effectively excited by noise which is
not pulselike or quasi-periodic. A similar model can be used, however, with the exci-
tation being a noise source.
We have seen that speech can be thought of as a convolution of two or more
continuous-time signals. In general, the Fourier transforms of the separate components
69
of speech are bandlimited (with the exception of the impulse train). Thus it is reasonable
to assume that the samples of the speech waveform can be obtained from discrete con-
volutions of the samples of the individual components. Thus, a reasonable model for
the samples of a segment of voiced speech is
s(n) = v(n) 0 g(n) (9 p(n),
where p(n) is a sequence that is approximately of the form
p(n) 6(n-rn p),
r
where n corresponds to the "pitch" of the sound.
4.2 SHORT-TIME TRANSFORM
Speech production can be modeled as we have just discussed, but implicit in our dis-
cussion was the fact that the character of the speech waveform changes as time proceeds.
That is, human speech is a string of sounds that are continuously changing. Thus the
model that we have discussed must be time-variant in the sense that both the excitation
function and the vocal-tract impulse response change as a function of time.
The fact that the character of the speech waveform changes with time, together with
the fact that for reasonably good quality speech we need at least 10,000 samples per sec-
ond to represent the waveform, requires that we adjust our notions about Fourier anal-
ysis. We find that it is neither possible or desirable to speak of the Fourier transform
of even an entire sentence, let alone the transform of longer segments. Rather, the
notion of a short-time Fourier transform is more appropriate.
Suppose we are interested in a segment of the speech waveform in the vicinity of
n =. (The time origin is arbitrary.) We define the short-time z-transform as
L- ý ,
S(z, ) = I s(n+g) w(n) z-n (104)
n=0
In Eq. 104, w(n) is a "window" containing L nonzero saniples. We "view" the speechsignal through this window by changing the parameter g, in order to move the segment
of interest under the window. The short-time Fourier transform of the sampled speech
is obtained by letthig z = e , that is,
L-1V -jwnS(eJ, g) = s(n+g) w(n) e . (105)
n=O
Since S(e J, 6) is really just the transform of a sequence of finite length, all of the prop-
erties of Fourier transforms of such sequences apply to (105). We note that the
70
sequence s(n+r) w(n) can be recovered by using the inverse Fourier transform relation
s(n+9•) w(n) \Sej, g' e 'n dw. (106)
In speech analysis, the length of the window is usually comparable to the duration of
the shortest speech sounds. We note that S(eJW, 6) is also equal to the convolution of the
transform of s(n+g) with the transform of w(n). Therefore if one requires good fre-
A quency resolution, longer windows may be required. This is the case in the echo
removal and detection applications for which we shall see that the window must be quite
long compar,• with the temporal detaid of the signal. When we are interested in speech
analysis, however, we find that we must make the window only a few pitch periods long
so that the character of the speech signal remains esse,.... Ily the same within the win-
dow. Typically, the duration of the window in this case is equal to ý number of samples
equivalent to 20-,*0 asec of speech. At a 10-kHz sampling rate this i' from 200 to 400
samples.
For speech analyisis, we have found the so-cailed Hamming window to be quite useful,
because of its good frequency resolution propertiec. The discrete Hamming window is
W(1 wll -f -Cos n)- 0 -<- n < L
(107)0 elsewhere
For echo removal and detection, we shall see in Section V that a truncated exponential
window of the form
w(n) a n 0 n<L
I 0 elsewhere
has properties that are very much suited to that application.
4.3 S1IORT..TNME COMPLEX CEPSTRUM OF SPEECHWe have seen that speech may be modeled as a convolution if we consider short seg-
ments of the waveform. We have also introduced the notion of a short-time transform
with an appropriate window. We shall now discuss how short-time !ransforms may be
used to obtain a short-time realization of homomorphic deconvolution.
Let us consider a segment of voiced speech that is multiplied by a window; that is,
wees<.i) w(n) = [vg(n)® ofpan+s) w(n), (108)
where vg(n) = v(n) & g(n). (We could, of course, assume without loss of generality that
the segment of interest occurs at 6 = 0.) In practice, the component vg(n) is of finite dura-tion, and we shall assume that it remains the same over the entire window. We observe
that the sequence whose values are s(n+g) w(n) for 0 4 n < L, however, is not strictly a
71
convolution. It should be clear that this is so beca'Ase the "tails" of previous periodsoverlap into the winoow and the "1tails"t of periods within the window are truncated by the
window. Nevertheless, under appropriate conditions this sequence will be approximately
a convolution so that the short-time transform for such a segment will be approximately
the product of the transform of vg(n) and the transform of a segment of p(n). This is
best illustrated as follows.
Let us assume that vg has a duration of not more than two pitch periods. We shall
assume that the window varies slowly so that w(n) = w(n+2n p), where n is the pitch
period. Then we may write Eq. 108 as
s(n+9) w(n) = vg(n) ® pw(n, ) + e(n, t),
where e(n, g) accounts for the overlaps at the beginning and the end of the window, and
Pw (n, ) = p(n+9) w(n).
Thus we have assumed that the window is such that vg remains essentially the same
across the whole window, with the pitch pulses being the only part that is weighted by the
window. If we evaluate the short-time transform we obtain
S(e3 'O, g) VG(e j) Pw(eJ), g) + E(eJ4, g).
Therefore we see that only if E(eJw, ý) is negligible, is S(ej", 9) simply a product. If it
is true that E(ejW, t) is negligible, however, we see that
S(ejWo, t) = log [S(e3o, ol)= log [P w(eJ". o)] + log [VG(e3 'w)].
Windows that taper to zero at the ends are quite effective in minimizing the end
effects.
We define the short-time complex cepstruri to be
I 1]ejwn Z As(n, 2r : r log [ S(ejr , g) ] d = pw(n, ý) + vg(n).
Thus the short-time complex cepstrum is the sum of w(n, ý), which contains the pitch
information in the interval spanned by the window, and the complex cepstrum of vg. The
latter can actually be tho,,ght of as the sum of two components, one that is due to the
vocal tract and the other to the glottal pulse. The impulse response of the vocal tract
is minimum phase; however, the glottal pulse is not. Therefore vg(n) will be nonzero
for both positive and negative values of n. Nevertheless, it is clear from the aperiodic
nature of vg, that the component vg(n) will tend to be concentrated around n = 0 and will
be bounded by
Iv^(n) 14 A II for all n.
72
On the other hand, p w(n, 4) is an impulse train which in general will give an impulse
train in the complex cepstrum, with the impulses occurring at multiples of the pitch
period. Thus we have a situation in which the complex cepstrum can be divided into two
regions, each corresponding to a different component of the speech signal. To recover
the component pw(n, g), we zero those samples of the complex cepstrum for InrI <np, and
then employ the inverse of the characteristic system to obtain p w(n,t,%. To recover vg,
we replace with zero those samples for InI >- np, and transform the result with the
inverse system. Examples of this are given in section 4. 4.
If the speech segment under consideration is unvoiced, the short-tine complex cep-
strum does not have the impulse train component as in the voiced speech. Thus, the
complex cepstrum can be used as a pitch detector and voiced/unvoiced detector for
vocoder and speech analysis applications. NoUll 1,12 has shown thai the cepstrum is
very successful in this application. (The cepstrum as defined by others is essei.. ially
the even part of the complex cepstrum.) It appears that the complex cepstrum may offer
advantages over the cepstrum, since it is ,ossble to actually extract the component
Pw (n, g) so that we obtain information about variations in pitch across the window, rather
than an average pitch period as is obts' - -d :n the cepstrum. The reason for this is that
if the pitch is not constant, the impulsep iW th,. epstrum and complex cepstrum eitiier
become smeared out around some averaztg f,. ch period, or impulses appear at longer
times than the fundamental period. The 4'ollowing examples illustrate thi8 point.
4.4 EXAMPLES
We shall first consider two examples. We shall consider the recovery of pitch, and
then the recovery of the component vg, which contains the impulse response of the vocal
tract and the glottal pulse.
As an example of the recovery of pitch, let us consider the segment of the vowel "ah"
as in father, shown in Fig. 27a. This waveform was sampled at a 10-kHz rate, and in
Fig. 27a the samples have been connected by straight lines to form a smooth curve.
(This is the case in all of the curves that we show in this section.) Note that the pitch
is quite constant in the interval shown.
In Fig. 27c is shown the complex cepstrum for the segment of Fig. 27a when weighted
by the Hamming window of Eq. 107, where L = 256 (25. 6 msec). Since the Hamming
window is very small in value at its ends, it tends to minimize the error caused by over-
lap. We note significant peaks in the complex cepstrum at appi oximately ±5 msec
(fli) samples). This is the period of the waveform in Fig. 27a. We also note that the
complex cepstcum is relatively small for values beyond 50 samples. Figure 27b shows
the output of the inverse characteristic system after having replaced the samples for
In I< 40 with zero. We note that the impulses ara spaced at the pitch period and
are weighted by the shape of the Hamming window.
As an examole of what happens wnen the pitch varies across the window, we
73
replaced the sampieo of Fig. 27c with zero for Inl > 40. Thus the pitch was removed,and the resulting output of the inverse characteristic system was a pulse containing
the glottal pulse and vocal tract information. This pulse is shown in Fig. 29c. This
pulse was used in the computer to synthesize a rnew waveform in which the spacing
of impulses in p(n) was alternating between 35 samples and 40 samples. This
waveform is shown in Fig. 28a. (Such pitch fluctuations have been reported by1'
Noll. 1
(a)
(b)
IW(c)I I -I I
-100 -50 0 50 100 n
Fig. 27. Pitch-extraction example. (a) Segment of the vowel "ah." (b) Theoutput for a long-pass system. (c) Complex cepst-um for Hammingwindow.
The complex cepstrum for Fig. 28a weighted by a Hamming window with L256 samples is shown in Fig. 2bc. This time, we note that there are significantpeaks at n = -75, -40, +35, and +75. The values of the complex cepstrum were
replaced by zero for Inm < 32 and the resulting output of the inverse characte.ristic
system is shown in Fig. 28b. It is clear that in this case, the output of the over-
all syrtem shows very clearly how the pitch changes in this time intervalr, On the
other hand, things are niot so clear in the complex cepstrum. In the (,:,en part of
74
the complex cepstrum, that is, the cepstrum, we would find peaks at n C 35, +40,
Sand +74. Thus we would probably have to be content with ar average constant pitch
over this interval, since the cepstrum does no tell in what order the long and
short periods occur.
: (a)
tV(b)
(c)
SIl I I
-100 -50 0 50 lOO
Fig. 28. Pitch extraction example. (a) Synthetic speech with fluctuating pa,,h.(b) The output for a long-pass system. (c) Complex cepstrum fo.,Hamming window.
The first two examples indicate that the use of the complex cepstrum may be
desirable in situations in which we are interested in very accurate and synchronous
pitch extraction. This remark is based, however, on only limited experimentation
and can o-ly be verified through more extensive effort i4 this direction.
As a third example, we note that in obtaining the waveform of Fig. 28a we
recovered a pulse from the complex cepstrum of Fig. 27c. This pulse is showu
in Fig. 29c. The original waveform is repeated in F'g. 29a with expanded time and
amplitude scales. To show that this pulse, together with pitch information for the
original waveform, is sufficient to recover the waveform, we synthesized the wave-
form of Fig. 29b, using pitch informantion obtained from Fig. 27b. We note that the
75
!
(b)
0 50 100 150 200
Fig. 29. Example of speech deconvolution, (a) Original speech (same asFig. 27a). (b) Synthesized speech using the pulse in (c)and pitchinformation in Fig. 27b. (c) Pulse obtained from the cepstrumof Fig. 27c, using the values for Inl 40. (Note: sampling rateis 10 kHz.)
waveforms (a) and (b) are not exactly the same, but they are quite similar in mostdetails. From this example and Fig. 27b, it seems clear that the assumptionsemployed in section 4. 3 are justified in this case. That is, the component vg isnot affected to a great extent by the weighting, while the pitch impulses retain theshape of the window.
76
V. APPLICATIONS TO ECHO REMOVAL AND DETECTION
5.1 A SIMPLE EXAMPLE
We have presented the theoretical and computational details of homomorphic decon-
volution. Now we shall be concerned with the application of these techniques to the pro-
cessing of signals containing echoes. To illustrate our approach, let us consider a
simple example. We can easily obtain analytical results for this example, and we shall
also present computational results for comparison.
Suppose we have a sequence x whose values are
x(n) = s(n) + as(n-n= s(n) ® p(n), (108)
where the sequence s has values
s (n) = nan 0 -< n< M
= 0 elsewhere.
Furthermore, 0 < a < 1, and
p(n) = 6(n) + a6(n-nl).
The z-transform of Eq. 108 is
X(z) = S(z) (I+az (109)
where S(z) can be shown to be
az- I (1-a-aMz-M MaMz-MS-z) - - I7 "
(1-az ) I -az
If M is large, aM approaches zero, so that for large M
S(z) az 1(1-az-)
The logarithm of X(z) is
X(z) = log [S(z)] + log [l+azi ,
which can be written
A .- 1 1 -nlX(z) - loga+ 2 l - og -az-+log i+az l (110)
Using the Laurent series expansion of the lopnrithm, we can write Eq. 110
,'7
.. Zt#)& ~ ~*4~J~tAEM&.J~A .t~ ~ ~ W,~ * ~t 4~fl t~t ~
' I"
0 512 0 80 512
(a) a (• ) (b) x(n) s(n) + -. 5s(n-80)
(C) log JXWk) (d) ARG [X(k)) 12
0 1024 0 80 160 240 320 512
(e) arg [X(k)) b2k!2048 M) (n)
Fig. 30. Waveform and transform for a simple example.
78
• - . --
A+ 2am -m -m+l a m -mn1)X(z) log gz- a + l ( I--z . (Il
M= I M= I
-1If we choose the contour of integration to be the unit circle, and remove the term log z
by shifting the sequence one place to the left, we obtain for the complex cepstrum
( 2ak + k+ 1 1-(n) (loga) 6(n) + 6 (n-k) + -1 - 6(n-knl). (112)
k= I k=I
From Eq. 112 we see that the contribution 'ue to p has samnles spaced at intervals
of n1 while the samples due the sequence s have unit spacing. Clearly both approach
zero faster than-L, but since the samples due to p have greater spacing, the part duen'
to s will occupy primarily the "short-time" region while the part due to the echoes will
be in the "long time" region.
This example was actually computed as discussed in Section In for a = .96, a = . 5,
M= 800, and n 1 = 80. The value of N for the DFT was 2048. The sequences s and xare shown in Fig. 30a and 30b, respectively. In Fig. 30c, we show the samples of thereal part of Eq. I10 for z = eJW. That is, Fig. 30c is
-j -4- k -j -&1- nlk
log IX(k)I =loga-Zlog l-ae N +log -l+ae N
In Fig. 30d we show the principal value of the phase of X(k). This includes the linear
phase component, as we can see, since
and the net number of positive and negative jumps between k = 0 and k = N2 - 1 is zero.
Figure 30e shows the phase curve after adding the corrections involved in removing the
discontin-,.. -es and rotating the sequence. Figure 30f shows the complex cepstrum for
this example. We note that the part attributable to s(n) is primarily concentrated between
n = 0 and n = 80, while the impulses attributable to the echo occur at n = 80, 160, etc.
We should point out that after rotating to the left one sample, the input sequence is mini-
mum phase, so that (n) = 0 n <0.
Let us now consider how wu might choose a frequency-invariant linear system to
recover each of the components s and p from the complex cepstrum. From Fig. 30b,
we see that to recover the sequence s, we must remove the impulses caused by p. One
way of doing this is to form
9()= I(n) ^X(n),
where f(n) is as in Fig. 31a, and nc < 80. This type of linear system is appropriately
79
termed a short-pass system. With this system, we would remove all of the contribu-
tion caused by p, and part of that caused by s. The analogy to lowpass filtering should
n n
I(n)
n 2n 3n n
/(n)
(c)nC n
Fig. 31. Frequency-invariant linear systems for (a), (b) echoremoval, and (c) detection.
be clear. If we consider Fig. 30c and 30e, we sae that both the log magnitude and the
phase are made up of the sum of a slowly varying component caused by s, and a more
rapidly varying component caused by p. In attempting to recover s, we must remove
the rapidly varying components while leaving the slowly varying component relatively
unaltered. Alternatively, if we have accurate knowledge of the echo time (80 samples
in this case), a comb system such as that shown in Fig. 31b would remove the contribu-
tion from p and would retain almost all of the part from s. In either case, operating
on 9(n) with the inverse system should produce an approximation to s(n). The compu-
tational example was carried out for these choices and the results are shown in Fig. 32.
In Fig. 32c we show the output for the comb system with nc = 80. Clearly, it is not pos-
sible to distinguish Fig. 32c from Fig. 30a. Similarly, we chose nc = 79 in the short-
ps .s system and obtained the output shown in Fig. 32d. Again, Fig. 32d is not
80
iIX
0 80 512 0 80 512
(a) Long-pass: nc-79 (b) Long-pass: nc-16
0) 80 160 512 0 80 160 512
(c) Comb: nc-80 (d) Short-pass: nc -79
0 16 80 512
(e) Short-pass: n -16
Fig. 32. Output waveforms for the example of Fig. 30.
"4
81
distinguishable from Fig. 30a; however, in Fig. 32c we have shown the output when the
short-pass system was used with nc = 16. A look at Fig. 30f shows that much of'•(n)
has been deleted by the linear system, and therefore we should not expect the output to
look exactly like s(n). The fact that the shape is the same is the result of the simplicity
of the distribution of zeros of S(z), and the fact that s is minimum-phase.
We shall say more about the implications of minimum phase in this example, but first
let us consider the problem of recovering the sequence p. From the complex cepstrum
in Fig. 30f it is clear that if we choose the linear system of Fig. 31c, with nc <80, most
of "(n) will be deleted, and we shall retain the part attributable to p. Such a system
could be appropriately called a long-pass system. In Fig. 32a we show the output of the
inverse system for a long-pass linear system with nc = 79. We note that very clearly
we have recovered a sequence whose values are very close to p(n). In Fig. 32b we show
the output for n. = 16. In this case, we have retained a significant part of •(n), and thus
the impulses of p are convolved with the filtered s(n). Since most of the significant part
of A(n) was removed, however, the part of the output from s(n) is very small.
We have previously noted that the input sequence was minimum-phase. Thus the
recursive algorithm will give the same result as that obtained by using the integral
expressions on the unit circle. The recursion formula for the inverse system is
y(n) = e9 (O) n = 0
n-]
= y(0) 9(n) + 1 - 9(k) y(n-k) n > 0,
k=OAA
where y(n) = i(n) •x(n).
The recursive expression helps us to understand the appearances of each of the out-
puts in Fig. 32. For example, in both Fig. 32a and 32b, A(0) = 0. Since
y(O) = e(0,
we see that since y is minimum phase and y(0) = 0, then y(0) = 1. Similarly, we see
that if 9(n) = 0 foro0 < n<nc, then y(n) = 0 for 0 <n<nc. This is shown by both
Fig. 32a, where nc = 79, and 32b, where nc = 16. We note that this explains why the
contribution attributable to s(n) in Fig. 32b begins 16 samples from each impulse.
In Fig. 32c, 32d, and 32e, we recall that 9(n) = X(n) for 0 _< n <nc. Thus from the
recursion formula, we see that y(n) = x(n) for 0 _< n < n (within computational accuracy).Ac
Since (n) = (n) 0 <n 4 nc in each case, we see thaty(n) = s(n) for0 _< n<nc. BecauseAs(n) is very small for n >, 80, we see that y(n) 2: s(n) for n >_ nc in Fig. 32c, and that y(n)
should be a better approximation in Fig. 32d, since the comb system retains most of
2(n). In Fig. 32e, we note that y(n) = s(n) 0 _< n < 16, but for n > 16, the output decays
much too fast. This can clearly be accounted for by the recursion formula.
This example has illustrated many of the important concepts in the use of
82
homomorphic deconvolution in processing signals containing echoes. We have seen that
it is possible to remove the echo from the sequence x by using a short-pass or comb
linear system. Furthermore, we have seen that it is possible to recover the impulse
train by using a long-pass system. We note that this impulse train can be used in echo
detection, and we shall see that it has advantages over using either the cepstrum or
complex cepstrum for this purpose. It should be clear that the characteristics of the
complex cepstrum of the impulse train are of primary importance in choosing the linear
system. Thus, we shall next study this question in detail.
5.2 COMPLEX CEPSTRUM OF AN IMPULSE TRAIN
An impulse train is defined as a sequence in which the nonzero samples are spaced
at intervals that are greater than one. An example of such a sequence is the sequence
" whose values are
M-1
p(n) = 6(n) + ak 6 (n-nk). (113)
k=1
Sequer.ces of this kind can be used to represent signals that contain echoes or reverbera-
tions. For example, the sequence whose values are
M-1
x(n) = s(n) + I aks(n-nk),
k=l
t can be represented as the convolution
x x=s p,.
where the values of p are given by (113), and s iL. a sequence whose values are spaced
at unit intervals. The complex cepstrum of x has values
A A Ax(n) = s(n) + A(n).
A.We have seen that even if the sequence is of finite length, the complex cepstrum s is ingeneral of infinite extent. Most of the energy in s is concentrated, however, in aninterval of the same order of magnitude as the duration of s. We sww in the example
that the impulse train
p(n) = 6(n) + a6(n-nl)
has a complex cepstrum
A ~ k
p(n) H) 6(n-knl).
k= 1
83
IThis component may be clearly seen in Fig. 30d. Thus, in this simple case, ve see
that the complex cepstrum is also an impulse train with samples spaced at int mrvals of
n We have also seen that the two sequences s and p may occupy essentially different
intervals so that it is possible to recover either s or p by using very simple frequency-
invariant systems. Since we must have some knowledge of the complex cepstrum of the
impulse train in order to choose the linear system, we shall consider in detail the prop-
erties of the complex cepstrum of an impulse train.
Let us consider an impulse train such as Eq. 113. The corresponding z-transform
is
M-1 _n
P(z) = 1 + akZ (114)
k= 1
In general, the analytical determination of the complex cepstrum corresponding to (114)
is quite difficult because it is quite hard to determine the zeros of the right-hand side
of (114). In the special case wherein the impulses are equally spaced, however, it is
possible to give a general result. Consider an impulse train
M-1
p(n) = ak6(n-kno).
k=0
Let us define a sequence q having values
q(n) =a 0-.< n< 4M-ln
= 0 elsewhere.
Thus we see that
p~ = q nn= O,n ... (M-l)n°
= 0 elsewhere.
The z-transform of p isM-lzn k kz
M-1 -kn M-1 Zn) no)P~z} = I a kZ o o=I a
k=0 k=0
Thus it is clear that p will be given by
A (n) (_n 0, *n 0 , *2nl 0 ,...
= C elsewhere.
84
That is, for impulse trains with equal spacing, the complex cepstrum is also an impulse
train with the same spacing. We note that even though p is a fin'te-length impulse train,
Sis in general of infinite duration.
If the spacing of the samples is not uniform, it is not possible to use the argument
above. Since we are not able to give a general result, we shall illustrate the range of
possibilities with several examples.
Examplel1
Let nk= kn and ak= ak in Eq. 114, so that P(z) can be written
M-1x. -knP(z) a ) az o.(115)
/..k=0
By a simple manipulation of (115), we can obtain the more convenient form
1 -a Mz-nP(z) = (1 16)
( -~~o)I -azn 0
The logarithm of P(z) is
P(z) = log (1-aMz n) -log (l-azo) ,
[ and if I, I < 1, we may write
A ' k -kn 0-1 k-~P(z) = Mk-ka-I -R - I -'a z 0.
k=1 k=l
(Note that such an expansion implies that the region of convergence includes the unit
circle.) Therefore, the complex cepstrum is
Co k 00 kMS(n) 6(n-kno) - 6(n-kMno). (i i 7)
k=1 k=l
A
The two sequences making up p are shown in Fig. 33 for M = 3. Thus, we see that if
the echoes are exponentially decreasing in amplitude and also equally spaced, the
complex cepstrum is an impulse train with the same spacing. We also note that p
is minimum-phase for Ia! < 1, and therefore ^(n) =0 for n < 0.
85
.$ i
f6
ý(n)
;((a)
a
r2 log/ i3/
'-0/
? 9 0 n n - -no 2n , 4 n . 5 .0 L, 6/ 7n °
3
1o)a-a
-4n lg 2(nI",. T 3/3
S-2n, n, +no J 3(n, •n0) n I
S-O2/2
(b)
Fig. 33. (a) Complex cepstrum for Eq. 118.(b) Complex cepstrum for Eq. 118.
Example 2
Let us suppose that
-nI-no -2nI -3n -nP(z) + az + apz (118)
where 10 1 > I and I=I < 1. Equation 118 may bt itten
P(z) = z -n1(lI+az-n°1- 0(l+P- z 2n
and the logarithm of P(z) is therefore
A -2n 1 ( -n.-n ) (-+nzZnl)2n1P(z) = log P + log z + log + +az + log.." (119)
If we assume that the region of convergence contains the unit circle, we may expand (119)
as
86
A-2nl 00• ( kl • -(n +no)k • 00-k 2nlk
P(z) = log p + log z + ÷ + (H 1)k+1 -- z nk= I k= I
If we also assume that the term, log z is removed in computing the phase curve, we
see that p(n) can be written
00oC -kk~l a k+li
P(n) (logP) 6(n/ + ()k - (n- -kno)+ I (- -- 6(n+k2n)k=1 k= 1
The sequence is depicted in Fig. 33b.
We note that in this case tht sequence p is nonminimum-phase after a shift left of
2nI samples, and therefore P(n) # 0 for n < 0. We note, however, that p is again an
impulse train with spacing related to the spacing of the sequence p. Clearly, it would
be difficult to detect all echoes in this case by using only the impulses in the complex
cepstrum. If, however, we use a long-pass system that replaces the samples in the
interval -2nI < n < n + no with zero, and then transform the result with the inverse
system, we shall obtain an output placing in evidence all of the samples of the
sequence p.
Example 3
Let P(z) be
-n I -n2
1 2P(z) a 1=az + a 2z ( 120)
where n 2 > n 1 . The logarithm of P(z) is
A -n! -n•P(z) = log 1+aIV +aZ2 z .
A
If we assume that the region of convergence of P(z) coutains the anit circle and that
-jwn -jwnl.max aI e +a 2 e < 1,
IT <W<Ir
then we may write
k kA o Z. +a aZ')~
Plz) = (1- k1 /(121)
k=1
(We shall say more about this restriction on P(z) after this example.) The binomrial
term may be expanded as
87
-nl n k k k-r kn rIaZ +az 2) 1 k- a az,
r=0
where the quantity (k) is the binomial coefficient
(k-r) I r I
Thus we can write (121)
A 0 k+1-1 k(k k-rr-(kn1 -rn2+rn I(-1) r I ak zk() H 1 2k=I r=O
and the complex cepstrum is therefore
co kA =+1 (_-rk+lk1 /rP(n) (-) k a a26(n-kn rn 1rn2)
k-1 r=O
This expression places in evidence several important facts. First, we note that P(n)
0 n < 0. This is a result of our assumption regarding aI and a 2. Second, we can see
that the impulses occur at
n = (k-r)nI + rn 2 k= 1, 2, ... and r = 0, 1 ... k.
The values of A(n) are given in Table 1 for 1 4 k _< 6.
The most striking thing about Examnplcs 2 and 3 is how much more complicated the
complex cepstrum becomes when the impulse train is nonminimumn phase, or when the
impulses are not equally spaced.
In general, we must consider impulse trains of the form of Eq. 113, and clearly
if the impulse train is both not equally spaced and nonminimum-phase, the complex
cepstrum will have impuLses located throughout the range -,0 < n < -o. It is to ou Y
advantage if the impuls! in the complex cepstrum occur onl3 in the "long-time"
region. Since the spacing dt the impulses is not under our control, we can only
look at the possibility of nz'.iking the impulse train minimum-phase. With respect
to this question we can make some definite statements. We recall that exponential
weighting can be used to make a sequence minimum-phase. We also recall that if
x is a convolution, its values ar'! given by
x(n) = I s(k) p(n-k).
k
The exponentially weighted sequence can be written
88
nx(n) = ' s(k) Pn-kp(n-k).
k
That is, each member of the convolution is also exponentially weighted. Therefore,exponential weighting can be used to make the impulse train minimum-phase. This
Table 1. Locations and values of the impulses in the complex cepstrum ofp(n) =6(n) + al16(n-nl) + a28(n-n2).
k n p (n) k n p(n)
IMna 5 5n, a15/5
I n22 5 4nl+n2 14a2S232
2 2n1 -a12/2 5 3n1+2n 2a13a2
2 nl n2 -a a2 5 2+3n 2a 2a23
1212 1+n2 2n2 -a2212 5 nl+4n alU24
22 5 a1 4 2 1 2
"r3 3nI 01 /3 5 5n a /5
2 63 2nl+' 2 a2 6 6nl -a /6
3 nl+2n ala 2 6 5nl+n -0 5 a
3 3n2 2 3/3 6 4n,+2n2 -5a14a22/2
4 4nI -a 1 /4 6 3n +3n2 -ba 3a2 3/3
4 3n+n -3aa 6 2n,+4n -5a 2a 4/21 2 1 2 __ 1 2 -5 1 1 /
4 2n+n 2 -2a 12a2
2 6 n -aIa 25
If21 _- - _-2
4 n +3n -a a23 6 6n /6__ 1 2 1 222
4 4n 2 -a 2/4
will then insure that the part of the complex cepstrum attributable to p will occur onlyin the region n >- n1, I ,•re n1 is the spacing between the first and second impulses.
In general we are concerned with
M-1_nk
P(z) =1 + az ,k (122)L. kk=1
89
where nI < n2 < ... <rnmI.* Since exponential weighting effectively replaces ak in
Eq. 122 by P kak# it is advantageous to have at least a sufficient condition that these
quantities must satisfy in order that the sequence be minimum-phase. Such a condition
is easily obtained since the requirement of minimum piiase is equivalent to requiring
that
P(z) = log + M akZ- (123)
k=I I
have a power-series expansion converging in a region jz > a, where 0 < a < 1. A suf-
ficient condition for this to be true is
NM-1max a e < 1. (124)
k= I
The condition of (124) is satisfied if
M-1 4
(125)k=lI
Thus (125) constitutes a sufficient condition for the sequence corresponding to (122) to
be minimum-phase. If the ak do not satisfy (125), the sequence may still be minimum-
phase (as in Example I); however, in general we can always insure that the impulse
train is mini,.num-phase by choosing P so that
M- n
k=l
Since we do not generally know in advance the size of the ak, we must rely on approxi-
mate information about the shortest echo time, the spacings, and the relative sizes of
the echoes in order to choose the proper value for p.In conclusion, we note that if the nequence p is minimum-phase, then the complex
cepstrum may be obtained from the following expansion of Eq. 123:
A , ,k+l I)' -KrP(z)- H) L a r .k=I \r=0
In this case we must use the multinomial expansion for (MV arzr), and clearly the\r=0
result will be a rather complicated distribution of impulses in the complex cepstrum.
90 4
We can state, however, that there will be no impulses in the complex cepstrum for
n < n. Furthermore, the impulse sizes will approach zero for large n and will occur
at
M-1
n in.n- mknk'k=1
where the mik take on all positive integer values.
5.3 DISTORTED ECHOES
Up to this point, we have assumed that the echoes were exact replicas of some
sequence s. In practice, of course, we are interested in sampled continuous-time
a h(t)DELAY
alhl~t)OF T,
Sa2h2(t) DELAY20t) OF T"2 X(t)
I.I
DELAY
aM-1hM'l1t) OF TM- I
Fig. 34. Linear model for the production of echoes.
signals, and this assumption is generally not exactly correct. A more realistic model
for the generation of continuous time signals containing echoes is shown in Fig. 34.
The input s(t) is assumed to be bandlimited, and the hk(t) specify the continuous-
time impulse responses of M - I linear systems corresponding to M - 1 different
echo paths. Thus x(t) is bandlimited, and is given by
x(t) = s(t) @ [u 0 (t)+aIhI(t-TI)+... +aM_IhMl(t-TMl)].
If we sample x(t) (assuming that the Nyquist frequency is 1), it can be shown
that the samples of x(t) are given by
x(n) = s(n) @ p(n)
"where
91
_ _ _ _ _ _ _ _ _ _
p(n) = 6(n) + aIhA (n-nI) + ... + aMIiA M (n-nM-l). (126)
In Eq. 126, the nk are integers and
Tk - nk - Ak'
where
0 Ak <1.
Thus, the hA (n) are the samples of hk(t+Ak); that is,k
hA (n) = hk(n+Ak).k
The Fourier transform of the sequence p is
M-1
P(eJW) = 1 + I akHk(eJ) e - ,nk (127)
k=l
where HA k(eJ) is the Fourier transform of the samples of h(t+Ak), and
H en= Hk (e3•w) e .U~
Hk(eJl) is the Fourier transform of the samples of hk(t).
This discrete model accounts for the fact that each echo may be distorted by
its transmission path and also for the small shift encountered if the echo delay
time is not an integer multiple of the sampling period.
We recall that the analysis for multiple echoes was quite difficult, and compli-
cated expressions were obtained for the complex censtrum. We also saw, however,
that the simple case of a single echo illustrates most of the important concepts.
Thus, for clarity, let us consider the special case
P(e n = 1 + aH (e3 n e
where H (ej) is the transform of the sequence of samples of a continuous-time, band-
limited, impulse response h(t+A). If we assume that
max aHA(eJ,)I < 1,
then the logarithm of P(eJ) can be written
92
P(eJj)= log + aHl(e3c ego
H) k+l a-kH (e3() e 1 (128)k- IkIA
The condition on JaHA(ed) I can be satisfied for all minimum-phase systems and manynonminimum-phase sysiems by exponential weighting of x(n), since nnx(n) will have the
transform
X(P-leJw)= S(P-leJ) P(P-lej),
where
wh r P(P-1 eJ )= I+ a nl HA(P-- ejt) e -J wn.
Thus we must choose P so that
max a n HA(P_-eJ(e) < 1.
The complex cepstrum for (128) is given by
p(n) k1(-I) k I(kh(n-knd], (129)k= 1
where
(k)h(n) = j . L [HA(ein k e jw dco. (130)
We also note from (130) that (k)hA(n) satisfies
(k) j(n) = (k-I)lhA(n) @ h,(n)
where
( 0)h,(n) : 6(n).
We see that if A = 0 and hA(n) = 6(n), Eq. 129 reduces to
A -1 k+I ak SP(n) H) 111~ 6(n-kn 1), .4
k=1
93
as we would expect. If this is not the case, then the impulses in the complex cepstrumare dispersed by convolution with )hln). In general, if the systems H are
kwideband, then the corresponding impulse response, hA (n), will be of short duration. On
kthe other hand, if H (eJWn is narrow-band, the impulse response will not approach zero
Akvery rapidly. In the first case, the impulses in the complex cepstrum will be convolvedwith relatively sharp pulses, while in the latter case the impulse will be convolved withsequences that are rather broad and dispersed.
To see how this affects our results, let us consider two examples.
Example 4
Consider a single echo path for which h(n) = a n > 0, where 0 <a < 1. It can beshown thst
(k)h(n) = (n+k-1)I an n > 0.
n1 (k-1) I
Thus we see that (k)h(n) becomes increasingly spread, because of the successive con-
volutions, and thus the peaks at large values of knI will be considerably smeared
6(n)
a1h(n-n 1)
0 n, n
nln
0nI 2n, 3nI 4nI n
(b)
Fig. 35. Impulse train for distorted echoes. (a) Sequence p(n) = 6(n) + alh(n-nl)for h(n) = an n>-O. (b) Complex cepstrum for p(n).
94
4A
out. This is especially true if a is close to 1. Figure 35 shows a plot of p(n) in (a) and
p(n) in (b) for this example. (The samples have been connected by straight lines for ease
in plotting.)
Example 5
Let us consider a single distorted echo obtained by filtering with a system whose
system function is
jO sq(w)H(eJ) = e
where sq(w) is given by
sq(w) =1 0 < <
1- -W < /< 0,
and sq(w) sq(w+r2t) r = 0, ±1, ±2 ..... Such distortions can occur in acoustic-wave
reflections. 1 ' Thus we can write
k Jk0osq(w) -jwkn 1P(eJ= (-I) -E- e .
k=1IJo 0 ksq(Q)
It can be shown that the sequence whose transform is e has values
(k)h(n) -2 sin 0 (s . n) n2 0= wn s2• n
cos 0 k n= 0.0
The complex cepstrum p(n) is given byzJ00
A(n = -1k+1l k+ I(p~k (-I))h(n-knl)
k=1
In this case, the shape of the sequence (k)h(n) remains the same for all values of k, but
we note that the relative size of the values of the sequence depends on the phase angle
0 0. In this case, and in the previous example wherein a is close to 1, it may be quite
difficult to remove the echo, since it will require a comb system to remove the compo-k
nents S (k)h(n-knl) without significantly disturbing the part that is due to s(n).
5.4 LINEAR SYSTEMS FOR ECHO REMOVAL AND DETECTION
We have seen that a simple model for a signal containing eL .oes is the convolution
x = s 0 p, where p is an impulse train in which the samples are spaced at intervals
greater than one. In section 5. 1, we presented an example that indicated that the
95
components s and p may be recovered by using frequency-invariant linear systems of
the formIA Ay(n) = 1(n) x(n).
We have shown that the essential concepts of the simple example are true in general,
even thoug.i, in many cases, the complex cepstrum becomes quite complicated. We now
wish to clarify our definitions and terminology and discuss some of the details of using
frequency-invariant linear systems for echo removal and detection. 2We recall that in section 5. 1 we introduced the terms "short-pass," "long-pass," and
"comb" systems. Precise definitions of these systems will now be given.
An "ideal short-pass system" is a frequency-invariant linear system for which
f(n) = 1 -n <n <n+
"= 0 elsewhere,
where n and n+ are integers. Such a system is shown in Fig. 36a.
An "ideal long-pass system" is defined as
f(n) = 0 -n_<n <n+
= I elsewhere,
where n and n+ are integers. This system is shown in Fig. 36b.
An "ideal comb system" is defined as
Ank An1(n) = 0 nk -- <n<nk + k k 1, 2,
= I elsewhere.
In general, nk can be a positive or negative integer. Such a system is shown in Fig. 36c.A.We note that in computation, the z-transform is replaced by the FFT, and x is replaced
Aby the periodic sequence x. Thus for computation, i(n) is effectively periodic, although1! Awe only work with one period of the sequence x.
If N [the number of samples of X(eJW)] is fairly large, it is faster to c,'npute the
complex cepstrum and multiply by 2(n) than to do the equivalent convolution of the trans-
form of 1(n) with the transform of •. The ideal systems of Fig. 36 are easily realized
in the computer by replacing the contents of appropriate registers with zero. It is well
known that such sharp cutoff systems will produce ripples in the transform of y; there-
fore, in many cases, it may be preferable to use approximations to these ideal systems
which have smoother transitions between one and zero.
In considering the advantages of homomorphic deconvolution over linear inverse fil-
tering, the basic consideration is the amount of information about the signals which is
required to design the system. Clearly, in inverse filtering we must have a good
approximation to the signal that is to be removed. If we do know this signal, a scheme
96
that is equivalent to inverse filtering is subtracting its complex cepstrum from that of
the input. In this case we could recover the desired signal by using the system D- 1 . One
cannot do better than this. In choosing to remove the undesired component with a linear
S/(4.A~n)
-n- n
(b)
1(n)
An, An2 An 3
(c)
Fig. 36. Ideal frequency-invariant linear systems: (a) short-pass;(b) long-pass; (c) comb.
system, we can expect to obtain only an approximation to the desired signal. The advan-
tage of homomorphic deconvolution is that for signals that are convolutions of an impulse
train with a sequence having its samples spaced at unit intervals the complex cepstra of
the two components are somewhat separated "in time." In this case, only partial infor-
mation about the signals is required.
From the examples that we have given, we can see that it is possible to remove
either of the two components if we are given only partial information about the impulse
train. For example, if the impulse train is minimum-phase (or has been made minimum-
phase by exponential weighting), and we know the smallest echo delay, we can recover
the impulse train by using a long-pass system that is zero for all n less than the
97
!IJI
smallest echo time. We note that the impulses will be convolved with the sequence
Ys=D-[I 1].
The values of this sequence will normally be quite small if the smallest echo time is long
enough so that most of the energy of *(n) is removed by the long-pass system.
A similar method is suggested for recovering the signal that is convolved with the jimpulse train. If we use a short-pass system that is equal to 0 for all n greater than
the smallest echo time and equal to 1 for n less than that value, we shall remove the
component attributable to the impulse train completely. We shall also remove part of
the complex cepstrum of s. In most practical situations, that is, when the echoes
(a)
(b)
4
(c)
1(n)
-259 96 253 n
(d)
Fig. 37. Effect of a short-pass system. (a) Input sequence.(b) Output for short-pass system of (d). (c) Com-plex cepstrum for (a). (Note: all traces have thesame time origin as 1(n).)
981
overlap significantly with s, this method generally requires that we remove too much
of ý to obtain a very good approximation to s at the output of the inverse system DA simple example of this is given in Fig. 32e, where a short-pass system that was equal
to 1 only up to n = 16 was used to recover 3. Another example is given in Fig. 37. In
Fig. 37a we show a segment of a speech waveform (without echoes) which has Leen
weighted with a Hamming window. In Fig. 37c we show the complex cepstrum of this
segment, and in Fig. 37d we show a short-pass system. The output of system D for
this case is shown in Fig. 37b. (The origin of all traces is in line with that of f(n).) By
comparing Fig. 37a and 37b, we observe that the two are aLmost identical for values of
n < 50. (The time axis in all plots coincides with d.) For n > 50 we observe that
Fig. 37b differs significantly from the original waveform. That this is true in generalfor short-pass systems can be shown from our discussion in section 2. 7.
Recall that for finite-length sequences with no linear phase component
m. m1 0
X(z) =A I(1-akzl 1 0 _b kI).k== 1
Thus x(n) is zero outside the interval -mo0 4 n -n mi. Recall that x and y may be written
x = min x max
Y Ymin Ymax'
where
/X mi.n (n) = (n) n >_ 0
-0 n<O,
Xmax(n) =(n) n < 0
=0 n >O .
We also see that for short-pass systems, as in Fig. 37d,
ymin(n) = xmin(n) 0 4 n 4 nc
-0 n<00 xm (n) nc < n
min
and
-Ymaxln) = xmax(n) -mo 0 n - 0
= 0 elsewhere.
99
- -
!A
It is relatively easy to see from these results that
=0 n<mJ 0
y(n) =x(n) -mo n n4n c-mo
#x(n) n -m0 <n.
In the example of Fig. 1, ,m 0 = 259 and ri = 253. Since nc = 96, we see that the
result above predicts that y(n) will be different from x(n) for all n > -163. This is not
detectable visually, until approximately n= 0. This is so because for n>nc - Mo, Ymax(n)
is relatively small, and it is only for larger valaes of n that this error in ymax is
reflected in y(n).
In general, the short-pass system is not as useful in echo removal as we might hope,
since in cases wherein the echoes significantly overlap we are required to remove too
much of the complex cepstrum of the desired output.
The alternative to the short-pass system is the comb system. In this case, we need
much more information about the impulse train in order to choose the values of nk and
Ank. We do not need to know the sizes of the echoes but either we must know their loca-
tions or the locations of the impulses in the complex cepstrum. If the echoes have been
distorted by a linear system (as discussed in section 5. 3), we also need to know the
approximate duration of the impulse response in order to choose the A k. In general, we
do not have this much information about the impulse train. Since th ý significant peaks
that are due to the impulse train stand out, however, in the complex cepstrum, except
for very short echo delays, we may detect these peaks and set the parameters of
the comb system appropriately. This can be done in an in-line computation system if
a display screen, or other graphical output device, is available, and if the experi-
menter is capable of interacting with the program. ('his was done in much of the experi-
mental work wh:rch is reported here. It is also clearly possible to program relatively
simple algorithms for searching appropriate regions of the complex cepstrum for
peaks attributable to the impulse train. The information obtained from such algo-
rithms can then be used to set the parameters of the comb system.
In connection with such algorithms, it is worth pointing out that we have found that
the phase component of the sequence s normally contributes a larger compo. at to
the complex cepstrum than the log magnitude. Thus, in order to make the detection of
impulses in the complex cepstrum easier, it is desirable to use only the even part of
the complex cepstrum. If the impulse train is minimum-phase, the impulses in the yen
part will occur at the same locations as those in the complete complex cepstrum. Thus
detection of the impulses in the even part of the complex cepstrum for n > 0 is equivalent
to detection of the impulses in the complex cepstrum for minimum-phase sequences.
We have discussed the basic forms of frequency-invariant systems that can be used
in echo detection and echo removal. Although the short-pass ,3ystem seems to distort
100
the desired output too much, it is possible to use a sort of self-adjusting comb system
for echo removal. We shall discuss examples of echo removal and echo detection, using
the ideas just formulated.
5.5 SHORT-TIME ECHO REMOVAL
We have discussed the properties of the complex cepstrum of sequences of the form
x=s Op,
where s is a sequence with values spaced at unit intervals, and p is an impulse train.
We also showed how frequency-invariant linear systems can be used to recover eithers or p. Throughout all of the previous discussion, we have assumed that the sequence xwas of finite length and that we were able to compute the z-transform (or FFT) of the
entire sequence. In some applications, for example, removal of echoes from speech
signals, the duration of the signal and high sampling rate combine to give sequences
with a great many samples. To process such sequences all at once, we are required
to take the FFT of a long sequence. In turn, for efficient operation, the FFT would need
a large amount of high-speed memory (that is, core storage). Thus, we are led to
inquire into the possibility of processing such sequences in shorter segments, and then
somehow putting these segments back together to form the complete output sequence.
This can indeed be done and we shall give an analysis of this procedure.
We shall begin with some definitions. We define thie short-time z-transform of the
sequence x to be
L-1
X(tIz) = x(g,n) z-", (131)
n=O
where
x(g, n) = x(g'n).
The short-time z-transform with window w is defined to be
L-1
n---0Xw(•,z) = • x•n ~)zn 12
n=0
Clearly, X w,, z) is just the z-transform of a finite-length, weighted segment of the
sequence x, and the parameter j simply serves to specify which segment is under con-
sideration. Therefore, if w(n) 0 0 for 0 4 n < L, then
x(g+n) = w I y Xw(4,Z) z n- dz, (133)Zwjw(n) ý'
where the contour C may be the unit circle.
101
',p
In particular, if w is an exponential window
w(n) = a Onn<LL,
we define
L-n
X(9,z,a) x(t+n) a z-n
n=O 0
We see that IX(t, z, a) = X(9, z/a)
and
M(, Z, 1) M (, Z).-
With these definitions in mind, let us assume that the sequence x has values
M-1
x(n) = s(n) + M aks(n-nk), (134)
k= 1
where the sequence s is a sequence with unit spacing of samples (such as a speechwaveform). The short-time z-transform with exponential window is
4L-1 M-1 L-1
X(4,z, a)= s(t+n)anz-n + I ak I s(g+n-nk) anz-n. (135)n=O k=1 n=O
If we let q = n - n k we can write
L-1 - L-1 -nak I st•+n-nk) a nz-n = ak a nk LknI s(g+q) aqz-q. (136)
n=0 q= -n k
Through some simple manipulations, (136) may be written
L-i
ak I s(t+n-nk) anz-n akS(' z, a) ank z -nk + Ek(•, z, a),
k=O
where
L-1
S(9, z,a) s(t+n) anzn (137)
n=0
102
and
S -1
Ekz, a) aakaZnk s(+n) anz-n
-az-L • s(+Ln) anz- (138)nknk
Thus Eq. 135 may be written
X(9, z, al = S(6, z, a) P(z/a) + E(g, z, a), (139)
where
M-1 nk -n kP(z/a) 1+ I a ka z
k= 1
and
M-1
E(9,z,a) = Ek(0,z,a).
k=1
Let us pause and interpret (139). Suppose that a is finite length, and s(n) = 0 forn < 0. If g = 0, and if L is greater than the total length of the sequence x, the termsEk(g, z, a) will all vanish in (139). Under these conditions, we are transforming all ofthe sequence at once, and we should expect that
X(O, z, a) = X(z/a) = S(z/a) P(z/a), (140)
where X(z) is the z-transform of x. Thus, the term E(6, z, a) is appropriately termed"the error in X(g,z,a)." That is, it is the amount by which Eq. 139 fails to have theform of the right-hand side of Eq. 140. The reason for these errors can be seen fromEq. 138. Let us write (138) as
Ek(g, z, a) = F k(Fz, a) -a.z-LFk(g+L, z, a), (141)
where
"|nk -n- -1n
_ k(•, z, a) = aka z k s(g+n) a 'z-n. (142)n- -nk
103
The first term on the right in (141) is seen to be due to overlap of the kth echo from the
previous segment into the present segment, That is, the samples s(g+n-nk) 0 < n < nk
do not appear in the basic segment s(g+n) 0 4 n < L. The second term on the right in
(141) is due to samples that appear in the segment s(6+n) 0 _< n < L and do not appear in
the segment of the echo s(++n-nk) 0 _< n < L, We shall refer to Fk(0, z, a) as the error
at the beginning of the segment, and -a Lz-Fk(4+L, z, a) as the error at the end of the
segment. We note that, except for a delay and multiplication by a constant, the error
at the end of the segment corresponding to 9 = 0 is the negative of the error at the
beginning of the segment corresponding to 6 = 0 + L.
Now let us consider the logarithm of X(g, z,a). We define
X(g,z,a) = log [X(g,z,a)] =log [S(4,z,a)P(z/a)+E(l,za)], (143)
and the short-time complex cepstrum as
x n, a)X(4, eJ),a) ej*On dw. (144)
We can write Eq. 143
X(g,z,a) (log [S(•,z,a) + 3 + log [P(z/a)]. (145)P(z/a)
From the second term on the right in (145), we see that the short-tin-e complex cep-
strum can be thought of as containing the same impulse-train component as t•Ae complex
cepstrum of the entire sequence. This assumption implies certain restrictions on the
length of the window and the nature of the signal s. These restrictions will be discussed
below. If this component were remo-ved by a comb system, the short-time transform
of the output segment would be simply
E(g, z, a)Y(gz,a) = S(6, z,a) + . (146)
P(z/a)
The samples of this output segment are
y(g+n)an =' Y(g, za) e d 0 <. n < L. (147)
-nThe effect of the exponential weighting may be removed by multiplying by a-n to obtain
the sequence of samples which has the transform
E(6, z, 1)Y(M Z. 1) -= S(4',Z.1) + .(148)
P(z)
From (148) it is clear that in the interval 0 4 n < L, the output segment is the sum
of the desired output s(g+n) and a sequence related to the error in X(g, z, 1). To see
104
the nature of this term, it is helpful to consider some special cazes.
Example 6
Assume that M 2; that is, there is only a single echo. Then from Eq. 141 we haveEl (9, z.ll)
Y(4 z,1) -- S( ,z, E1i,) +,1
1 + a1Z
Thus the output segment has values
00
y(ý+n) = s(t+n) + I (-al1)r eI (+n-rnl),
r=-0
for n >_ 0. This example is illustrated in Fig. 8.
s( n +L )
L-nI L n
0 e(C+n)
S~L L+nj
(b)
I (-)r* (C+n- rn1)
nN 2n| 3n I
(c)
Fig. 38. Errors for short-time echo removal. (a) The sequence s(g+n). (b) Error
in the input for a single echo with a = 1/2, nI. (c) Error in the output.
105
105
Example 7
Assume that =a= k and = kn1 . Therefore, from Eq. 148,
M-1Z Ek(gz, 1)k=9
k=O
which can be written
M-I
Y(9, z,I) =S(g, z,l) + 1- z Ekl)
k= I
if aM is small. Thus the output segment has values
M-1y(g+n) P, s(g+n) + I [ek( g+n)-ae k( g+n-knl)].
k= 1
Using Eq. 141, we can write
M-1 M-1 M-I
SEk(4 z'l) = I Fk(0 ,z l) -z-L I Fk(6+L z, 1),
k= I k= 1 k= 1
and using Eq. 142 and some manipulation, we obtain
M--n -1-nI n-1 -nl - 21
1 -az1) Fki''zl - az s(g+n) z-n + a 2z s(g+n) z n
k= I n=-nI n:.-2nI.
M- (-~,-(M-2)nl-I n
+ ... +a z I s(g+n) z-nn=-(M-1)n I
Mn -Mn-a z s(g+n) z-. (149)
n=-MnI .+.
Careful examination of the terms in (149) shows that all of the terms on the right except
the last contribute to the error of y(g+n) only in the interval 0 4 n < nI, while the lastterm is quite small but does contribute over the interval 0 4< n < hinl, An identical
result holds for the error at the end of the segment so that the error term
106
M-1I [e ek( 4+n)_ae k(6+n-knl)]
k= I
will be approximately zero everywhere but in the intervals 0 4 n <nl, and L 4 n <L+ n1 . 4
An example of this is shown in Fig. 42f where the first two traces correspond to 0 4 n <L,
and the last two traces correspond to L 4 n < ZL.
Example 8
Let there be two echoes not necessarily equally spaced. Then, from Eq. 148,
E1( z, 1) + Ez(9, z,1Y(4',z,1) = S(g, z,1) +
-n -n2I + a Iz I+ a2 z
We could proceed as before. In this case, however, the equations become so complex as
to be relatively useless. We can see from Eq. 138 that the sequence corresponding to
EI(t,z,l) + E 2 (•,z,I)
will be nonzeru only in the intervals 0 4 n < n2 and L 4 n < L + n.. This sequence isS n ý-n I
then convolved with the sequence corresponding to I + az- + a z which wil.1 be
an impulse train.
The three previous examples have several things in common. We note that the term
M-IPtE('z'l) = I Ek(t' z' 1)k
is the amount by which X(6, z, 1) fails to be a product of S(g, z, 1) and P(z) the trans.-
form of the impulse train. We have called this the error in X(g, z, 1). Similarly, we
note that
E(t, z, l)
P(z)
is the amount by which Y(6, z, 1) fails to be the desired output S(g, z, 1). We shall refer
to this as the error in Y(t, z, 1). Note that we use the term "error$, in a slightly dif-
ferent sense in this case. The error in the output cmn be thought of as the error in the
input, passed through the inverse system for the impulse train that represents the
echoes. We have seen that the error in the output consists of two segments, one of whichis primarily in the interval 0 4 n < L, and a second segment which is in the region L4 n.
We have also seen that in general these errors tend to become small for large n. Fur-
thermore, the error in the interval L 4 n for the segment 0 is the negative of the
error in the interval 0 4 n < L for the segment 9 t + L. This fact will be used when
107
rI- *!--* -. ,*.---* .- *4~ A~
we discuss putting the output segments back together.
In obtaining the previous results it was assumed that the short-time complex cep-
strum can be thought of as containing a component that is equal to the complex cepstrum
of the impulse train that produces "he echoes. In order for this to be true, the length
of the segment L must be large compared with nrM-l, the longest echo time. Further-
more, the signal s must change character over the segment of interest. These points
are clear intuitively, but quantitative results appear to be difficult to formulate. Clearly,
we want the error segments in the input to be short compared with the total length L.Since the length of these error segments is equal to the longest echo delay, we require
L >)nM_1
This means that the corresponding errors in the output will be concentrated primarily
in the two intervals 0 4 n < L and L 4 n < 2L.
With respect to the character of the signal s, let us consider an example. Suppose
s is a sine wave so that
s(n) =sin (n) n > 0
=0 n<0.
If the sequence x has values
x(n) = s(n) + as(n-nl,
it is clear that all segments of x for which g > n1 will look just lilke a sine wave with
some phase shift. That is,
x(n) = sin n + a sin (n-nl)
= (I+acosnl) sinn- sin nI cosn n>n1
= A sin (n+O).
All periodic sequences suffer the same difficulty, as well as exponential sequences of
the form an n >_ 0. In all of these cases, the short-time complex cepstrum will not
exhibit an impulse train because of the echo. Speech signals, for example, change
character as time progresses. Clearly, the requirements on the character of the wave-
form and length of the segment L are interrelated. If we make the segment long enough,
almost any nonperiodic signal will change sufficiently across the segment.
The previous results were based on the 7.-transform, whereas we shall realize such
processing by using the FFT. This, of course, means that the short-time complex .ep-
strum that we compute will be aliased. znd will have values given by
^ N-i j 21 knx(4, n, a) X (4, n+ rN, a) = log [Xý(9, k, a)] e ,
r=-oo k=0
108
where N >, L, and
N-I 2w k-jn-kn(, k, a) = x(gnae N
n=T
The resulting output of the system D-1 after removing the impulse train caused by theechoes is approximately the exponentially weighted output
00•a anry(+n+rN).
If we unweight with a-n in the interval 0 n < N, we obtain approximately
y(g, n) = y(6, n) + a Ny(g, n+N) (ISO)
for 0 4 n < N. All indices are taken modulo N.
From (150) we see that if N = L, the error in y(g, n) is primarily at the beginning of
the segment, since it is composed primarily of the error in the beginning of y(g, n) plus
the error at the end of aNy(ý, n+N). If we choose N = ZL, the error in y(6, n) is approxi-
mately the error in y(g, n). Thus, in this case, we can apply most of the pre. 'it results
even though aliasing does occur.
0 L 2L 31 4L 5L 6L 71III I I I I I x
ADD
•=L I I
ADDC=2L I I I
ADD I
0 L 2L 3 41 5L 6L 71
,II I I I I I l•
Fig. 39. Correction method for short-time echo removal.
Having discussed the actual computation of the short-time complex cepstrum, and
resulting output segment, we are able to indicate how the output segments may be put
together to form the total output sequence. We have noted that if the errors in the ,.-
109
'4
output tend to approach zero, the error in the interval L < n < 2L for the segment =
is approximately the negative of the error in the interval 0 < n < L for the segment ius
io + L. This suggests that we can move the window along in jumps of L samples. Each
time we save the output for L 4 N < 2L from the segment • = so that it may be addedto the values in the interval 0 4< n < L of the segment 0 • + L. This method is illus-
trated in Fig. 39. We refer to this method as the "correction method."
As an alternative to the above scheme, we note that if the segment is quite long com-
pared to the longest echo time, the error is generally small over some interval
Lo n < L so that we may retain only this part which is relatively free of error. This
0 L 2L-Lo 3L-2L 4L-3LI I I I I 0
SAVE
DROP SAVE
2(L-L 0 DROP SAVE
C=3(L-L 0 DROP SAVE
0 L 2L-L 31-2L 4L-3LS1i I° I
Fig. 40. Overlap method of short-time echo removal.
scheme is shown in Fig. 40, and it is referred to as the "overlap method." Examples
indicating the feasibility of these schemes will be given.
5.6 REMOVAL OF ECHOES FROM SPEECH SIGNALS
We shall present some examples of the application of the previous results. We shall
show several examples of the removal of computer-simulated echoes from speech sig-
nals. In all figures, each trace in a given picture represents 1024 sa nples unless other-
wise noted. The consecutive traces represent consecutive 1024 sample segments of the
speech waveform. The entire waveform corresponds to the sentence, "A pot of tea helps
to pass the evening.," The sampling rate was 10 kHz. In all examples, we used an expo-
nential window with a = 0. 9987 and L = 2048. The value of N for the FFT was N = 4096.
The segment of speech was, therefore, augmented with 2048 zeros before transfor-
matron.
110
-
-___J-
"- -•-.. -
I 1 t •02(a) (b)
1. 1024 I 1024 A
(c) (d)
1024
(e)
Fig. 41. Short-time echo removal for an impulse train
p(n) = 6(n) + 3/4 6(n-500).
(a) Signal s(n). (First 4096 samples.)(b) Signal x(n) = s(n) @ p(n).(c) Complex cepstrum for the first 2048 samples of x.(d) Ouitput for the first 2048 samples.(e) Error for the second 2048 samples.
Ill
Example 9
The speech signal was convolved with
3p(n) = 6(n) +-! 6(n-500).
The first 4096 samples of the signal s and the sequence x are shown in Fig. 41a and
41b. The complex cepstrum for the segment corresponding to • 0 is shown in
Fig. 41c. We note the impulse appearing at n - 500. The result of removing the
impulses at n = 500, 1000, etc., using a comb system, and then transforming the
result with the system D-I is shown in Fig. 41d. The waveform in the interval
0 4 n < 2048 is indistinguishable by eye from the first two traces of Fig. 41 e. We also
note that the error at the end of the segment appears repeated as predicted by
Example 6. There is no error at the beginning of the segment, sine the speech signal
waa essentially zero for n < 0. The segment corresponding to • 2048, that is, the
second two traces in Fig. 41b, was processed in the same way, and the difference
between the resulting output and the original segment augmented with zeros (the error
in the output) is shown in Fig. 41e. We note that this is just the error in the output, and
we see clearly that the first two traces in Fig. 41e are approximately the negative of
the last two traces in Fig. 41d. If the last two traces of Fig. 41d were added to the first
two traces in Fig. 41e (that is, the segment for g = 2048), the error in the interval
0 _< n < L would be essentially eliminated. Similarly, we could save the last two traces
of Fig. 4e, to use as a correction for the next segment.
We also can see from Fig. 41e that the error is relatively small in the interval
1024 _< n < 2048. This suggests that we could also use L = N = 4096, and disregard the
first 1024 samplea in each output segment, as suggested by Fig. 40. In this case, we
would obtain 3072 sainpies per segment, rather than 2048 as in the correction method.
Example 10
in this case, the echoes were specified by
M-1
p(n) = 6(n-k500).
k=0
The first 4096 samples of the sequences s and x are shown in Fig. 42a and 42b. The
short-time complex cepstrum obtained from the first two traces of I g. 42b augmented
with 2048 zeros is shown in Fig. 42c. The impulses at n = 500, 1000, etc. , were
remoied with a comb system, and the resulting output is shown in Fig. 42d. The second
two traces in Fig. 42b were procebned similarly, and the difference between this output
ant' the lae• 2 traces of Fig. 42a augmented with 2048 zeros (the desired output) is shown
in Fig. 42f. We note that the error is concentrated in the interval 0 _< n < 500, as pre-
dicted by Example 7. We also see that the error in the interval 0 -< n < 2048 in Fig. 42f
is the Tiegativ? of the error in the interval 2048 4 n < 4096 in Fig. 42d.
1 11?
I 1024
(a) (b)
I 1024 1024
(c) Md
aim -
02 1024
(e) (f)
Fig. 42. Short-time echo removal for an impulse train
(a) Signal s:oz. (First .. sa.. i(b) Signal x(n) = s(n) (& p(n).
(c)•!, Comle .ep u fo the. fis 204 sam-pl-s ofx
(d) Output for *he first 2048 samples.(e) Output for th(b second 2048 samples.(f) Error for th( second 2048 samples.
113
Example 11
As a final example, the speech signal was convolved with
3 1p(n) = 6(n) + ÷ 6(n-320) + - 6(n-500).
Again, the first 2048 samples of the resulting sequence were augmented with 2048 zeros
and processed as before. The error for the first output segment is shown in Fig. 43a.
The error for the second segment (t = 2048) is shown in Fig. 43b. In Fig. 43c we show
the sum of the last two traces of Fig. 43a and the first two traces of Fig. 43b. We note
that the errors clearly tend to cancel. Significant error would remain in the second
segment, however. This error is primarily due to the fact that the window is not long
enough relative to the echo time. (To see the size of the error relative to the desired
output see Fig. 41a.)
AU of the previous examples have indicated that it is possible to put the output seg-
ments back together in a meaningful way either by using the correction method or the
overlap method. This was done, in fact, for several different variations of echo times,
and the resulting output speech was converted to analog form for listening. informal
listening tests showed that if a suitable value of L is chosen, echoes can be removed
from speech signals by using these techniques. The processed speech was slightly more
noisy than the input speech. This noise in the output is attributable to the fact that
there are small errors remaining in each segment, as in Fig. 43c.
5. 7 EFFECT OF ADDITIVE NOISE
The examples that we have shown were carried out on signals with a high signal-
to-noise ratio. Suppose that we have a sequence x which is of the form
x = s@ p + g,
where s is the desired signal, p is an impulse train, and g is an additive noise
sequence. The short-time transform of a segment of the sequence x is
X(t, z) = S(t,z) P(z) + E(4,z) + G(t,z),
where S(6, z) is the short-time transform of the signal, P(.) is a polynomial in z-
tf .t is the transform of the impulse train, E(6, z) is the error as defined in sec-
tion 5. 5, and G(t, z) is the short-time transform of the noise. If the noise level is
low, we may again assume that the short-time complex cepstrum has a component
that is due to P(z) which can be removed by a comb system. The output sequence
will then be of the form
E(4 z) + G(tz)Y(M,z) = S(M,z) + . (151)
P(z)
Thus, in addition to the errors previously discussed, we see that the noise in the
114
1024
(b)
(C)
Fig. 43. Short-time echo removal for the impulse train
p(n) = 6(n) + (3/4) 6(n-320) + (1/2) 6(11-500).(a) Error in the output for the first 2048 samples of x(n)=
s(n) 0 p(n). (s(n) is the same as in Fig. 41.)(b) Error in the output for the second 2048 samples of xi
15"F(g 4) Sumoof-thme sechon removao trcso(aanthe firsts trai
_ ~ traces of (b).
Fig. 44. Short-time echo detection
M-I
p(n) = (3/4)k 6(n-k832).
k=O
(a) Complex cepstrum of the second 2048 samples of x(n) •s(n) D p(n). (s(n) is as in Fig. 41a.)
(b) Output for R long-pass system.
p(n) = 6(n) + h(n-832)
h(n) = (3/4)n/4 n !1 0
=0 n< 0
(c) Complex cepstrum for the second 2048 samples of x.
(d) Output for a long-pass system.
p(n) = 6(n) + (3/4) 6(n-832) + (1/2) 6(n-1536).
(e) Complex cepstrum for the second 2048 samples of x.
(1) Output for a long-pass system.
116
116
I/
1024 I 1o24
(a) (b)
1024 1024(C) (d)
102 1024
(e) M)
117
------i- ---
input segment also appears in the output. We note that the noise is also effectivelyfiltered by the linear system corresponding to 1/P(z).
To test the effectiveness of short-time echo removal at low signal-to-noise ratios,
we added a white Gaussian noise sequence to the sequence so p, where p was an
impulse train with equal spacing. Signal-to-noise ratios as low as 15 dB were used. The
sequences were processed as before, and the resulting outputs were converted to analog
form. informal listc'iing tests for the speech signals showed that in the examples, at
least, the echoes were removed and the noise level of the output was about the same as
that of the input. We should point out, however, that only limited experimentation was
done, and a clear understanding of the effect of additive noise is yet to be obtained.
5.8 DETECTION OF ECHOES
We have considered a signal containing echoes to be represented by a convolution of
a basic signal and an impulse train. We have also seen that the complex cepstrum of an
impulse train is itself an impulse train. We have shown how echoes may be removed
by using a linear frequency-invariant system (possibly on a short-time basis). In con-
clusion, let us consider the problem of detection of echoes, that is, recovery of the
impulse train p(n).
As might be expected, the problem of echo detection is not as difficult as the problem
of extraction of the signal. We have seen that in simple cases, the complex cepstrum,
cr its even part, may be used for echo detection. If the impulse train is not equally
spaced, however, the problem of determining the number and locations of all of the
echoes from the complex cepstrum becomes quite difficult. It is therefore interesting
to consider some examples of recovery of the impulse train.
If the impulse train is minimum-phase (or has been made minimum-phase by expo-
nential weighting), then we have seen that the impulses attributable to the echoes occur
only in the region n >_ ni, where n1 is the shortest echo delay. Thus a long-pass system
that multiplies by zero for all n < n1 can be used to recover an approximation to the
impulse train. The caiculations can be further simplified if the impulse train is
minimum-phase, since we can use the log magnitude alone to compute the even part
of the complex cepstrum. Then the long-pass system can be chosen to perform the
Hilbert transform operation on the impulse train, as well as remove most of the
minimum-phase part of the signal.
We have contended that the short-time complex cepstrum can be thought of as having
a componert caused by the impulse train. If this is so, then we aLo should be able
to carry out short-time echo detection. To show that this is true, let us consider
some examples.
Example 12
The speech signal was convolved with
118
M-1
p(n) (3 / 4)k 6(n-k832).
k=O
The input was exponentially weighted and tho even part of the short-time complex cep-
strum was computed by using only the log magnitude of the FFT. The length of each seg-
ment wab L = 4096 samples and N = 8192. In Fig. 44a we show the cepstrum for the
second segment (ý + 4096), Each trace corrn 1ponds to 1024 samples. Note the impulses
at 832, 1664,and so forth. After multiplying by 2 for n>-800 and by 0 for n < 800, the out-
put of the inverse system appears as in Fig. 44b. We have recovered a very good
approximation to p(ii).
Example 13
The speech signal was convolved with
p(n) - 6(n) + 3/4 6(n-832) +-T 6 (n-15 3 6 ).
As in the previous example, L = 4096 and N 8192. In Fig. 44e and 44f we show the cep-
strum and output for the second segment (• = 4096). Again, it Ž clear that the output is
a very good approximation to p(n).
Example 14
The speech signal was convolved with
p(n)= 6(n) + h(n-832),
where
h(n) (3/4) n >- 0
=0 n<0.
In Fig. 44c we show the cepstrum, and in Fig. 44d we show the output after processing
as in the other examples. We see that the impulses in the cepstrum are convolved with
the sequences (k)h(n), as discussed in section 5. 3. The output also shows that we have
recovered a good approximation to p(n). Note how the impulses are dispersed in the
cepstrum. This, of course, makes it even more difficult to detect the echoes in the
cepstrum.
119
IJ
"TI. CONCLUSION
6.1 SUMMARY
We have presented a new approach to separating convolved signals. A detailed anal-
ysis of homomorphic systems for deconvolution has been given, and we have shown how
such signal transformations may be realized by using a digital computer. As an appli-
cation we have considered the class of signals in which one of the members of the con-
volution is an impulse train. Although our examples have dealt primarily with speech
analysis and the removal of echoes from speech signals, it should be emphasized that
almost all of our results, particularly in Section II, apply to more general situations.
Therefore, it is felt that the point of view that is reflected in this work is important and
our results have demonstrated that homomorphic deconvolution may be a useful approach
in many interesting problems.
6.2 SUGGESTIONS FOR FUTURE RESEARCH
Although some interesting results have been obtained, there are still significant
questions that would be worthy of further investigation. For example, it is quite pos-
sible that other computational realizations can be obtained. This comment is prompted
by our observation in section 2. 7 that for sequences of length M, a total of M values
of the complex cepstrum suffice to completely determine the original sequence. For
nonminimum phase sequences, a direct method of computation of the necessary values
of the complex cepstrum which would avoid aliasing would be a worthwhile result.
Ano+her issue is the question of appropriate window functions to use in short-time
analysis. In speech analysis, this is an irrmportant consideration. It is also possible
that other weighting sequences besides the exponential can be found which would tend to
minimize the errors that we have discussed with respect to short-time echo removal.
In both speech analysis and echo removal, it would also be useful to have more general
results on the complex cepstrum of an impulse train v ith nonuniform spacing.
One of the issues that we have only touched upon is the effect of additive noise.Limited experimental results have been obtained, but adequate understanding of this
issue is a challenging problem.
As well as the issues relating to carrying out homomorphic deconvolution, it is of
interest to consider situations in which our techniques might be successfully applied.
The present work has shown that there are clearly advantages to homomorphic deconvo-
lution when one of the signals is an impulse train. Given the ease with which one can
think of signals of this class, it seems clear at this point that the techniques presented
here should rind application in many diverse areas. For example, it is possible
that homomorphic deconvolution may be used to obtain very accurate synchronous
pitch detection. Also, it appears that there are possibilities of application to seis-
rmic signals, for both dereverberation and detection. Still another area may be inprocessing underwater acoustical signals.
120
APPENDIX
Vector Space for Convolution
A. 1 DEFINITIONS
A. 1. 1 Field
A field consists of a set of objects called scalars (in our case numbers), together with25
two operations called addition and multiplication which satisfy the following conditions.
1 . To every pair of scalars a and b, there corresponds a scalar a + b called the
sum of a and b such that
(a) a+b=b+a
(b) a + (b+c) = (a+b) + c
(c) there is a unique zero scalar, such that a + 0 = a
(d) to every scalar a there corresponds a unique scalar -a such that a + (-a) = 0.
2. To every pair of scalars a and b there corresponds a scalar ab called the
product of a and b such that
(a) ab = ba
(b) a(bc) = (ab)c
(c) there exists a unique scalar 1 called one such that al = a,
(d) to every nonzero scalar a, there corresponds a unique scalar a such that-1
aa =1
(e) a(b+c) = ab + ac.
Examples of a field are the sets of real numbers, of rational numbers, and of com-
plex numbers, where addition and multiplication have their usual meaning.
A. 1.2 Vector Space
A vector space consists of a field of scalars, together with a collection of elements
called vectors having the following properties.2 5
1. To every pair of vectors x and y there corresponds a vector x + y called the
sum of x and y such that(a) addition is commutative, x + y = y + x(b) addition is associative, w + (x+y) = (w+x) + y
(c) there is a unique zero vector such that x + 0 = x
(d) there is a unique inverse vectot^ x + (-x) = 0.
2. To every scalar a and vector x there corresponds a vector ax such that
(a) a(bx) = (ab)x
(b) lx r x for every vector x
(c) a(x+y) = ax + ay
(d) (a+b)x = ax + bx.
We shall verify that convolution is an appropriate operation for vector addition, and
attempt to clarify the meaning of scalar multip)ication for convolutional vector spaces.
121
A. 2 CONVOLUTION AS VECTOR ADDITION
Our set of vectors is the set of all sequences whose z transforms exist and have
overlapping regions of convergence. The operation of convolution is
00
x 0 y- x(k) y(n-k). (A. 1)
k= -o
By a simple zhange of summation index, we see that convolution is commutative, that is,
00
x 8y= I y(k)x(n-k)y 0x.
k= -o
Similarly,
00 00 00 00
(W 00 X) 00 y w(k) x(m-k) y(n-m) I w(k) I x(m-k) y(n-m)m=-oo k=-oo k=-oo m=-oo
00 00
= w(k) I x(m) y(n-k-m) w 0 (x 0 y),
k=-oo m=-00
so that convolution is also associative. The zero vector is clearly the sequence 6 such
that
6(n) 0 n 0
=I n=0.
The inverse sequence for a sequence x is simply the inverse z transform of I /X(z),
where X(z) is the z transform of x.
A. 3 SCALAR MULTIPLICATION
We shall denote scalar multiplication by (b)x. To begin to see what scalar multipli-
cation means tor convolutional vector .paces, let us assume that b is an integer. Con-
ventionally we say that multiplication of a vector x by an integer is equivalent to adding
the vector to itself b times. This is a direct consequence of the postulates because, for
example,
2x = (l+1)x = x + x.
Thus in the case of convolution we say that (b x corresponds to the convolution of x
with itself b times. That is,
(b)x (b-1) X x,
122
Ir
where
(x)x =x
(O)x 6
and
(-I)x @x 6.
The set of all positive and negative integers does not constitute a field; however, the
set of all rational numbers does. In this case, it is only slightly more difficult to inter-
pret scalar multiplication. For example, suppose
y =(1/2) x.
A reasonable interpretation in this case is given by the expression
= 1y = y 0 y.
In general for rational scalars we can give the interpretation
Vb(b) (a)y xy- x,
where a and b are integers.
An alternative interpretation of the scalar multiplication results from consideration
(-,f the z transform. For example, the z transform of (b)x, where b is an integer, is
[X(z)lb. That is, X(z) raised to the bth power. In the second case, we note that for
y= x,
we obtain
[y(z)]b = [X(z)]a.
Alternatively, the function [X(z) ]a/b is normally defined by
[X z) I/b = expa log
where [X(z)Ia/b is clearly a multivalued function of z. in fact, we may define
b[X(z)b = exp{b log [X(z)] (A. 2)
for b a real or complex number.
It is evident from Eq. A. 2 that the definition of scalar multiplication is intimately
related to the proper definition of log [X(z)]. In order that jX(z)]a have a Laurent
123
expansion and thu- be thought of as a z transform, we require that [X(z)]a be single-
valued. This can be accomplished through the concept of the Riemamn surface.22 Under
the assuryiption that [X(z)]a uniquely defined, the verification of the conditions
regarding scalar multiplication involves only straightforward manipulations of powers
of the z transform.
It
Acknowledgment
This report represents only a part of an extremely enjoyable
and stimulating educational experience. For this experience, I am
indebted to many people.
In particular, I wish to thank Professor Alan V. Oppenheim. I
was fortunate to have been able to work closely with him during the
development of the present research, and our association has been
one of the highlights of my experience at the Massachusetts Insti-
tute of Technology. I am also grateful to Dr. T. G. Stockham, Jr.,
who assisted in the supervision of my thesis, but, more importantly,
contributed his enthusiastic support when it was needed most.
Thanks are also due to Professor Amar G. Bose, who also assisted
in the supervision of my thesis, and whose comments and personal
example have shown me the value of clarity of thought and expres-
sion.
I would like to express my appreciation to Professor Y. W. Lee
and Professor Henry J. Zimmermann, of the Research Laboratory
of Electronics, M. I. T. , for their generous sv.pport of my thesis
research. I would also like to thank Dr. I. L. Lebow of Lincoln
Laboratory, M. J. T. , for making it possible for me to use the
TX-2 computer.
124
References
Homon.-rphic Systems
1. A. V. Oppenheim, "Superposition in a Class of Nonlinear Systems," TechnicalReport 432, Research Laboratory of Electronics, M. I. T., Cambridge, Mass.,March 31, 1965.
2. A. V. Oppenheim. "Optimum Homomorphic Filters,r Quarterly Progress ReportNo. 77, Research Laboratory of Electronics, M. I. T., Cambridge, Mass., April 15,1965, pp. 248-260.
3. A. V. Oppenheim, "Nonlinear Filtering of Convolved Signals,' Quarterly ProgressReport No. 80, Research Laboratory of Electronics, M. L T. , January 15, 1966,pp. 168-175.
Deconvolution
4. T. G. Kincaid, "Time Domain Analysis of Impulse Response Trains," TechnicalReport 445, Research Laborntory of Electronics, M. I. T., Cambridge, Mass.,May 31, 1967.
5. James R. Reeder and S. S. Wolff, "Fourier Techniques for the Resolution of Over.lapping Waveforms," Proceedings of the Third Annual Allerton Conference on Cir-cuit and System Theory, University of Illinois, Urbana, Illinois, 1965, pp. 310-319.
6. E. A. Robinson, "Predictivo Decomposition of Time Series with Application toSeismic Exploration," Geophys. Vol. XXXII, No. 3, pp. 418-484, June 1967.
7. M. M. Sondhi, "An Adaptive Echo Canceller," Bell System Tech. J., Vol. XLVI,pp. 497-511, March 1967.
S8. Tzay Y. Young, "Epoch Detection - A Method for Resolving Overlapping Signals,"Bell System Tech. J. , Vol. XLIV, pp. 401-426, March 1965.
9. R. W. Schafer, "Echo Removal by Generalized Linear Filtering," 1967 NEREMRecord, Boston, 1967.
The Cepstrum
10. B. P. Bogert, M. J. R. Healy, and J. W. Tukey, "The Quefrency Alanysis ofTime Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum andSaphe Cracking," Proceedings of the Symposium on Time Series Analysis, editedby M. Rosenblatt (John Wiley and Sons, Inc. , New York, 1963), Chap. 15, pp. 209-243.
11. B. P. Bogert and J. F. Ossanna, "The Heuristics of Cepstrum Analysis of a Sta-tionary Complex Echoed Gaussian Signal in Stationary Gaussian Noise," IEEE Trans-actions )n Information Theory, Vol. IT-1Z, No. 3, pp. 273-380, July 1966.
12. A. M. Noll, "Short-Time Spectrum and 'Cepstrum' Techniques for Vocal-PitchDetection," J.Acoust. Soc. Am. 36, 296-302 (February 1964).
13. A. M. Noll, "Cepstrum Pitch Determination," J. Acoust. Soc. Am. 41, 293-309 (February 1967).
Fast Fourier Transform
14. J. W. Cooley and J. W. Tklkey, "An Algorithm for the Machine Calculation of ComplexFourier Series," Math. Computation, Vol. 19, No. 90, pp. 297-301, April 1965.
125
I -
~g1I;
I
15. W. M. Gentleman and G. Sande, "Fast Fourier Transforms- for Fun and Profit,"19b6 Fall Joint Computer Conference, AFIPS Proc., Vol. 29, pp. 563-578 (SpartanBooks, Washington, D.C., 1966).
16. 'Special Issue on Fast Fourier Transform," IEEE Trans. on Audio and Electro-acoustics, Vol. AU-15, Jure 1967.
17. T. G. 3tockham, "High-Speed Convolution and Correlation," 1966 Spring Joint Com-puter Conference, AFXPS Proc. , Vol. 28, pp. 229-233 (Spartan Books, Washington,D.C., 1966).
Speech Analysis
18. J. L. Flanagan, Speech Analysis Synthesis and Perception (Academic Press, Inc.,New York, 1965).
19. A. V. Oppenheim and R. W, Schafer, "Homomorphic Analysis of Speech," IEEETrans. on Audio and Electroacoustics, Vol. AU-16, pp. 221-226, June 1968.
20. A. V. Oppenheim, "A Speech Analysis-Synthesis System Based on HomomorphicFiltering," J. Acoust. Soc. Am. 45, 458-465 (February 1969).
Linear Systems
21. H. S. Carslaw, Introduction to the TheorX of Fourier's Series and Integrals (Dover
Publications, Inc., New York, 3rd edition, 1952).
22. R. V. Churchill, Compley Variables and Applications (McGraw-Hill Book Co., Inc.,New York, 1960).
23. H. Freeman, Discrete-Time Systems (John Wiley and Sons, Inc. . New York, 1965).
24. E•. A. Guillemin, Theory of Linear Physical Systems (John Wiley and Sons, Inc.,New York, 1963). -
25. P. R. Halmos, Finite-Dimensional Vector S2aces (D. Van Nostrand Co., Inc.,Princeton, N. d., 1958).
26. E. I. Jury, Theo and Application of the z-Transiurm Method (John Wiley andSons, Inc., New Yor-R-, 1964).
27. W. M. Siebert, "Notes for M. I. T, Course 6. 05," !967 (unpublished).
126
I
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Echo R,rnoval by Discrete Generalized Linear Filtering
4 TDESCAPTIE NOTES (TYPeO e1tenrt.anjlusivif.dte.)Teclieal Report
A4t THORIS) (FI,15 namie, middle initile, last nome)
Ronald W. Schafer
REPORT DATE 7a. TOTAL NO OF PAGES 17b NO OF REFiSFebruary 28, 1969 162
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11 495.TIACT
A new approach to separating convolved signals, referred to as homomorphic decon-olution, is presented. The class of systems considered in this report is a member oflarger class called homomorphic systems, which are characterized by a generalized
rinciple of superposition that is analogous to the principle of superposition for linearsystems.Adetailed analysis based on the z-transform in given for discrete-time systems i
of this class. The realization of such systems using a digital computer is also dis-cussed in detail. Such computational realizations are made pocsible through the appli-'ation of high-speed Fourier analysis techniques.
As a particular example, the method is applied to the separation of the compo-ents of a convolution in which one of the components is an impulse train. This classf signals is representative of many interesting signal-analysis and signal-processingroblems such as speech analysis and echo removal and detection. It is shown thatomornorphic deconvolution is a useful approach to either removal or detection ofchoe s.
FORM 1 (PAGE UNCLASSIFIEDI NOV ASamfcaio
Securtyt Clas,ffecitlon
- '-An -"--
1 44 lt * I 14, I',
homomorphic ystmt,,'rns
deconvolution
echo removal
speech analysis
complex cepstru m
Fast Fourier Transform I
i IFE
II
IIII
III
UNCASIFIE
